Recent zbMATH articles in MSC 53https://zbmath.org/atom/cc/532023-12-07T16:00:11.105023ZUnknown authorWerkzeugBook review of: F. Galaz-García (ed.) et al., Mexican mathematicians in the world. Trends and recent contributionshttps://zbmath.org/1522.000132023-12-07T16:00:11.105023Z"Arroyo-Rabasa, Adolfo"https://zbmath.org/authors/?q=ai:arroyo-rabasa.adolfo"Simental, José"https://zbmath.org/authors/?q=ai:simental.jose-eReview of [Zbl 1495.53005].Book review of: S. P. Kerckhoff (ed.) et al., The collected works of William P. Thurston with commentaryhttps://zbmath.org/1522.000272023-12-07T16:00:11.105023Z"Bonahon, Francis"https://zbmath.org/authors/?q=ai:bonahon.francisReview of [Zbl 07581719; Zbl 07581721; Zbl 1512.57001; Zbl 1507.57005].Book review of: I. Stavrov, Curvature of space and time, with an introduction to geometric analysishttps://zbmath.org/1522.001202023-12-07T16:00:11.105023Z"Suceavă, Bogdan D."https://zbmath.org/authors/?q=ai:suceava.bogdan-dragosReview of [Zbl 1472.83001].Bourbaki seminar. Volume 2021/2022. Exposés 1181--1196https://zbmath.org/1522.001912023-12-07T16:00:11.105023ZPublisher's description: This 73rd volume of the Bourbaki Seminar gathers the texts of the sixteen lectures delivered during the year 2021/2022: surface groups in lattices of Lie groups, non-density of integral points and variations of Hodge structures, Ricci flow and diffeomorphisms of 3-manifolds, structure of limit spaces of non-collapsed manifolds, classification of joinings, Shelah's conjecture and Johnson's theorem, high-dimensional expander graphs, non-linear spectral gaps and applications, local marked length spectrum rigidity, subconvexity problem for \(L\)-functions, non-linear Schrödinger equation, Kannan-Lovász-Simonovits's conjecture, binary additive problems over finite fields, crystalline measures, \(K(\pi,1)\) conjecture for affine Artin groups, sets with no three terms in arithmetic progression.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Kassel, Fanny}, Surface groups in lattices of semisimple Lie groups, 1-72, Exp. No. 1181 [Zbl 07722664]
\textit{Maculan, Marco}, Nondensity of integral points and variations of Hodge structures, 73-119, Exp. No. 1182 [Zbl 1522.11067]
\textit{Maillot, Sylvain}, Ricci flow and diffeomorphisms of 3-manifolds, 121-131, Exp. No. 1183 [Zbl 07722666]
\textit{Mondello, Ilaria}, Structure of limit spaces of non collapsed manifolds with Ricci curvature bounded from below, 133-179, Exp. No. 1184 [Zbl 07722667]
\textit{Aka, Menny}, Joinings classification and applications, 181-245, Exp. No. 1185 [Zbl 07722668]
\textit{Anscombe, Sylvy}, Shelah's conjecture and Johnson's theorem, 247-279, Exp. No. 1186 [Zbl 07722669]
\textit{Wagner, Uli}, High-dimensional expanders, 281-294, Exp. No. 1187 [Zbl 1519.05144]
\textit{Eskenazis, Alexandros}, Average distortion embeddings, nonlinear spectral gaps, and a metric John theorem, 295-333, Exp. No. 1188 [Zbl 1521.30072]
\textit{Hamenstädt, Ursula}, Local marked length spectrum rigidity, 335-352, Exp. No. 1189 [Zbl 07722672]
\textit{Michel, Philippe}, Recent progress on the subconvexity problem, 353-401, Exp. No. 1190 [Zbl 1522.11039]
\textit{Perelman, Galina}, Finite time blow up for the compressible fluids and for the energy supercritical defocusing nonlinear Schrödinger equation, 403-432, Exp. No. 1191 [Zbl 1522.35373]
\textit{Aubrun, Guillaume}, Towards Kannan-Lovász-Simonovits' conjecture, 433-451, Exp. No. 1192 [Zbl 07722675]
\textit{Kowalski, Emmanuel}, Binary additive problems for polynomials over finite fields, 453-478, Exp. No. 1193 [Zbl 07722676]
\textit{Meyer, Yves}, Crystalline measures and applications, 479-494, Exp. No. 1194 [Zbl 1522.42012]
\textit{Haettel, Thomas}, The \(K(\pi,1)\) conjecture for affine Artin groups, 495-546, Exp. No. 1195 [Zbl 1521.20074]
\textit{Peluse, Sarah}, Recent progress on bounds for sets with no three terms in arithmetic progression, 547-581, Exp. No. 1196 [Zbl 07722679]Curvature on graphs via equilibrium measureshttps://zbmath.org/1522.050732023-12-07T16:00:11.105023Z"Steinerberger, Stefan"https://zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by \(K > 0\) have diameter bounded by \(\operatorname{diam}(G) \leq 2/K\) (a Bonnet-Myers theorem), that \(\operatorname{diam}(G) = 2/K\) implies that \(G\) has constant curvature (a Cheng theorem) and that there is a spectral gap \(\lambda_1 \geq K/(2n)\) (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin-Lu-Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.
{{\copyright} 2023 Wiley Periodicals LLC.}Heat flow and concentration of measure on directed graphs with a lower Ricci curvature boundhttps://zbmath.org/1522.051662023-12-07T16:00:11.105023Z"Ozawa, Ryunosuke"https://zbmath.org/authors/?q=ai:ozawa.ryunosuke"Sakurai, Yohei"https://zbmath.org/authors/?q=ai:sakurai.yohei"Yamada, Taiki"https://zbmath.org/authors/?q=ai:yamada.taikiSummary: In a previous work [Calc. Var. Partial Differ. Equ. 59, No. 4, Paper No. 142, 39 p. (2020; Zbl 1446.05040)], the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower Ricci curvature bound via a gradient estimate for the heat semigroup, and a transportation inequality along the heat flow. As an application, we will conclude a concentration of measure inequality for directed graphs of positive Ricci curvature.Geodesic orbit and naturally reductive nilmanifolds associated with graphshttps://zbmath.org/1522.051942023-12-07T16:00:11.105023Z"Nikolayevsky, Y."https://zbmath.org/authors/?q=ai:nikolaevskij.yu-a|nikolayevsky.yuriSummary: We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.Bounded degree cosystolic expanders of every dimensionhttps://zbmath.org/1522.052692023-12-07T16:00:11.105023Z"Evra, Shai"https://zbmath.org/authors/?q=ai:evra.shai"Kaufman, Tali"https://zbmath.org/authors/?q=ai:kaufman.taliSummary: In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes yields for every dimension an infinite family of bounded degree complexes with the topological overlap property. This answers an open question raised by \textit{M. Gromov} [Geom. Funct. Anal. 20, No. 2, 416--526 (2010; Zbl 1251.05039)].Kähler graphs whose principal graphs are of Cartesian product typehttps://zbmath.org/1522.053972023-12-07T16:00:11.105023Z"Adachi, Toshiaki"https://zbmath.org/authors/?q=ai:adachi.toshiakiSummary: We consider that vertex-transitive normal Kähler graphs are candidates of discrete models of homogeneous Kähler manifolds by comparing their adjacency and spherical mean operators. To provide examples of such Kähler graphs we define some product operations of graphs. We study connectivity and bipartiteness of their principal and auxiliary graphs, and investigate commutativity of their adjacency operators. We then get normal Kähler graphs whose principal and auxiliary graphs are connected. Being inspired with examples of such graphs of product type, we give a way of constructing vertex-transitive normal Kähler graphs whose vertex-cardinality are multiples of 4 and whose principal and auxiliary degrees \(d^{(p)}\), \(d^{(a)}\) are odd with \(d^{(p)} - d^{(a)} \equiv 2 \pmod 4\).
For the entire collection see [Zbl 1508.53008].Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifoldshttps://zbmath.org/1522.054472023-12-07T16:00:11.105023Z"van der Hoorn, Pim"https://zbmath.org/authors/?q=ai:van-der-hoorn.pim"Lippner, Gabor"https://zbmath.org/authors/?q=ai:lippner.gabor"Trugenberger, Carlo"https://zbmath.org/authors/?q=ai:trugenberger.carlo-a"Krioukov, Dmitri"https://zbmath.org/authors/?q=ai:krioukov.dmitriSummary: Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier-Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.Fourfolds of Weil type and the spinor maphttps://zbmath.org/1522.140542023-12-07T16:00:11.105023Z"van Geemen, Bert"https://zbmath.org/authors/?q=ai:van-geemen.bertThe article under reviewed consists of an exposition of some recent results proved by \textit{K. G. O'Grady} [Int. Math. Res. Not. 2021, No. 16, 12356--12419 (2021; Zbl 1490.53068)] and \textit{E. Markman} [J. Eur. Math. Soc. (JEMS) 25, No. 1, 231--321 (2023; Zbl 07665043)] on certain families of abelian fourfolds admitting specific Hodge classes.
Let us describe briefly the construction. Starting with a \(\mathbb{Z}\)-module \(W\) of rank~\(4\), \(V:=W\oplus W^*\) is naturally endowed with a bilinear form of signature \((4,4)\). It is easily shown (Lemma~0.4) that orthogonal complex structures (\textit{i.e.} complex structures that preserve the bilinear form) are in one-to-one correspondence with the set of maximally isotropic subspaces \(Z\subset V_\mathbb{C}\) such that \(V_\mathbb{C}=Z\oplus \overline{Z}\). These subspaces form a \(6\)-dimensional complex manifold that can be realized as a quadric \(Q^+\subset \mathbb{P}S^+_\mathbb{C}\) where \(S^+\) is a lattice of rank~\(8\) endowed with a bilinear form \((\cdot,\cdot)\). The open subset
\[
\Omega^+:=\left\{ \ell\in\mathbb{P}S^+_\mathbb{C}\mid (\ell,\ell)=0\ \text{and}\ (\ell,\bar{\ell})>0\right\}
\]
is then a parameter space for a family of tori \(\mathcal{T}_\ell:=V_\mathbb{C}/(Z_\ell+V)\). A construction (using the Clifford algebra and the spinor representation) enables us to define a class \(c_s\in\bigwedge^4 V\) that can be interpreted as a cohomology class on the torus \(\mathcal{T}_\ell\). The compatibility condition with the complex structure \(Z_\ell\) reads like this: the class \(c_s\in\mathrm{H}^4(\mathcal{T}_\ell,\mathbb{Z})\) is a Hodge class (of type \((2,2)\)) if and only if \((s,\ell)=0\). Playing with another element \(h\), it is possible to construct a family of abelian fourfolds with very interesting properties: this is the content of Theorem~4.6.
Theorem. Let \(h\) and \(s\) such that the restriction of \((\cdot,\cdot)\) to the plane \(\langle h,s \rangle\) is positive definite and define
\[
\Omega^+_{h,s}:=\left\{ \ell\in\mathbb{P}S^+_\mathbb{C}\mid (\ell,\ell)=(\ell,h)=(\ell,s)=0\ \text{and}\ (\ell,\bar{\ell})>0\right\}.
\]
This \(4\)-dimensional manifold parametrizes a complete family of polarized abelian fourfold of Weil type and such that the Cayley class does not lie on the line generated by \(\omega_\ell^2\) where \(\omega_\ell\) is the above-mentioned polarization.
Being of Weil type means that these abelian fourfolds are endowed with complex multiplication \(K\subset\mathrm{End}(\mathcal{T}_\ell)\) where \(K=\mathbb{Q}(\sqrt{-d})\) where \(d:=(h,h)\cdot(s,s)>0\) (see Section~4.2 for the precise definition). Finally when the \(\mathbb{Z}\)-module \(W\) we started with satisfies \(W=\mathrm{H}^2(A,\mathbb{Z})\) with \(A\) an abelian surface, the whole construction can be interpreted in the framework of moduli spaces of sheaves on this abelian surface as in [\textit{S. Mukai}, Am. J. Math. 117, No. 6, 1627--1644 (1995; Zbl 0871.14025); \textit{K. Yoshioka}, Math. Ann. 321, No. 4, 817--884 (2001; Zbl 1066.14013)]. This makes the link with the works Markman and O'Grady alluded to above.
This article gives a \textit{pedestrian} (as said in the abstract) way to get to Markman and O'Grady's results: the notions are elementary and the computations are made as explicit as possible. It is very well written and will be useful for everyone willing to enter this circle of ideas.
Reviewer: Benoît Claudon (Rennes)Symplectic involutions of \(K3^{[n]}\) type and Kummer \(n\) type manifoldshttps://zbmath.org/1522.140572023-12-07T16:00:11.105023Z"Kamenova, Ljudmila"https://zbmath.org/authors/?q=ai:kamenova.ljudmila"Mongardi, Giovanni"https://zbmath.org/authors/?q=ai:mongardi.giovanni"Oblomkov, Alexei"https://zbmath.org/authors/?q=ai:oblomkov.alexei-aA complex hyperkähler manifold \(X\) is called of \textit{\({K3}^{[n]}\)-type} if it is deformation equivalent to the punctual Hilbert scheme \(\operatorname{Hilb}^n(S)\) of some \(K3\) surface \(S\). For such hyperkähler manifolds \(X\), the authors study fixed loci \(F\) for symplectic involutions.
The main result is, roughly speaking, that such fixed loci look like in the standard situation, for punctual Hilbert schemes and involution induced from a symplectic involution on the \(K3\) surface. In particular, \(F \) is a union of hyperkähler manifolds of \({K3}^{[m]}\)-type with \(m\leq n/2\), together with isolated points, and there are precise formulas for the number of connected components.
This generalizes, for example, results of Nikulin on the number of fixed points on \(K3\) surfaces [\textit{V. V. Nikulin}, Tr. Mosk. Mat. O.-va 38, 75--137 (1979; Zbl 0433.14024)] and Mongardi on fixed loci on Hilbert squares of \(K3\) surface [\textit{G. Mongardi}, Cent. Eur. J. Math. 10, No. 4, 1472--1485 (2012; Zbl 1284.14052)]. An analogous statement holds when the standard model is a Kummer variety of some abelian surface rather then a punctual Hilbert scheme of a \(K3\) surface.
The proof relies on the Global Torelli Theorem, deformation theory, and lattice theory.
Reviewer: Stefan Schröer (Düsseldorf)Quadratic Lie algebras with 2-plectic structureshttps://zbmath.org/1522.170282023-12-07T16:00:11.105023Z"Bajo, Ignacio"https://zbmath.org/authors/?q=ai:bajo.ignacio"Benayadi, Saïd"https://zbmath.org/authors/?q=ai:benayadi.saidSummary: We study the existence of 2-plectic structures on Lie algebras which admit an \(\text{ad} \)-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.A new look at Lie algebrashttps://zbmath.org/1522.170372023-12-07T16:00:11.105023Z"Dobrogowska, Alina"https://zbmath.org/authors/?q=ai:dobrogowska.alina"Jakimowicz, Grzegorz"https://zbmath.org/authors/?q=ai:jakimowicz.grzegorzSummary: We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair \((F, v)\) consisting of a linear mapping \(F \in \operatorname{End}(V)\) with an eigenvector \(v\). This pair allows to build a Lie bracket on a dual space to a linear space \(V\). The Lie algebra obtained in this way is solvable. In particular, when \(F\) is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie-Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.Extension of topological groupoids and Hurewicz morphismshttps://zbmath.org/1522.180162023-12-07T16:00:11.105023Z"Chatterjee, Saikat"https://zbmath.org/authors/?q=ai:chatterjee.saikat"Koushik, Praphulla"https://zbmath.org/authors/?q=ai:koushik.praphullaThis paper introduces the notion of Morita equivalence of topological groupoid extensions, investigating its relation with a gerbe over a topological stacck. The paper offers a topological version of the authors' previous work [Bull. Sci. Math. 163, Article ID 102886, 30 p. (2020; Zbl 1453.14003)] within the smooth setup. The most distinctive feature in the topological setup is the representability of a morphism of stacks obtained from a topoloigcal space (Lemma 2.10), which in turn shows the existence of a pair of atlases related by a map of local sections (Lemma 2.11).
A Serre fibration is a continuous map of topological spaces that has homotopy lifting property with respect to CW complexes, while a Hurewicz fibration enjoys homotopy lifting property with respect to all topological spaces. A Hurewicz (respectively, Serre) stack is a topological stack that is representable to a topological groupoid
\[
\left[ X_{1}\rightrightarrows X_{0}\right]
\]
whose source map is a Hurewicz (respectively, Serre) morphism of stacks. The notions of Serre and Hurewicz stacks were introduced in [\textit{B. Noohi}, Adv. Math. 252, 612--640 (2014; Zbl 1310.14006)]. The main contributions of this paper are the introduction of a Hurewicz (respectively, Serre) gerbe and the observation on its relation with a Hurewicz (respectively, Serre) morphism of stacks.
Reviewer: Hirokazu Nishimura (Tsukuba)Classification and decomposition of quaternionic projective transformationshttps://zbmath.org/1522.201972023-12-07T16:00:11.105023Z"Dutta, Sandipan"https://zbmath.org/authors/?q=ai:dutta.sandipan"Gongopadhyay, Krishnendu"https://zbmath.org/authors/?q=ai:gongopadhyay.krishnendu"Lohan, Tejbir"https://zbmath.org/authors/?q=ai:lohan.tejbirSummary: We consider the projective linear group \(\mathrm{PSL}(3, \mathbb{H})\). We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of \(\mathrm{PSL}(3, \mathbb{H})\). We further decompose elements of \(\mathrm{SL}(3, \mathbb{H})\) as products of simple elements, where an element \(g\) in \(\mathrm{SL}(3, \mathbb{H})\) is called \textit{simple} if it is conjugate to an element of \(\mathrm{SL}(3, \mathbb{R})\). We have also revisited real projective transformations and following Goldman's ideas, have offered a complete classification for elements of \(\mathrm{SL}(3, \mathbb{R})\).Injective metrics on buildings and symmetric spaceshttps://zbmath.org/1522.201982023-12-07T16:00:11.105023Z"Haettel, Thomas"https://zbmath.org/authors/?q=ai:haettel.thomasSummary: In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise \(\ell^\infty\) metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an \(\ell^\infty\) metric on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of \(\operatorname{GL}(n,\mathbb{R})\) as the space of all norms on \(\mathbb{R}^n\). The only exception is the special linear group: if \(n=3\) or \(n \geqslant 5\) and \(\mathbb{K}\) is a local field, we show that \(\operatorname{SL}(n,\mathbb{K})\) does not act properly and coboundedly on an injective metric space.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean spacehttps://zbmath.org/1522.260122023-12-07T16:00:11.105023Z"Cabré, Xavier"https://zbmath.org/authors/?q=ai:cabre.xavier"Miraglio, Pietro"https://zbmath.org/authors/?q=ai:miraglio.pietroIn this paper, the authors delve into the examination of Hardy-Sobolev inequalities on hypersurfaces of \(\mathbb R^{n+1}\). These inequalities encompass a mean curvature term and feature universal constants that remain unaffected by the specific hypersurface under consideration. Initially, the authors focus their attention on the renowned Sobolev inequality established by \textit{J. H. Michael} and \textit{L. M. Simon} [Commun. Pure Appl. Math. 26, 361--379 (1973; Zbl 0256.53006)] and \textit{W. K. Allard} [Ann. Math. (2) 95, 417--491 (1972; Zbl 0252.49028)] within the context of codimension-one. Building upon their insights while simplifying their expositions, the authors furnish a succinct and easily comprehensible proof of this inequality.
\par Subsequently, the authors introduce two novel Hardy inequalities on hypersurfaces. One of these inequalities arises from its utility in the realm of regularity theory concerning stable solutions to semilinear elliptic equations. The other, proven through the application of a ``ground state'' substitution, serves as an enhancement to Hardy inequality of \textit{G. Carron} [J. Math. Pures Appl. (9) 76, No. 10 (1997; Zbl 0886.58111)]. Employing the same methodology, the authors also attain an improved version of the Hardy or Hardy-Poincaré inequality.
Reviewer: Petr Gurka (Praha)A remark on the quaternionic Monge-Ampère equation on foliated manifoldshttps://zbmath.org/1522.300262023-12-07T16:00:11.105023Z"Gentili, Giovanni"https://zbmath.org/authors/?q=ai:gentili.giovanni"Vezzoni, Luigi"https://zbmath.org/authors/?q=ai:vezzoni.luigiThe quaternionic Monge-Ampère equation was introduced on hypercomplex manifolds by \textit{S. Alesker} and \textit{M. Verbitsky} [J. Geom. Anal. 16, No. 3, 375--399 (2006; Zbl 1106.32023)]. They conjectured that the equation should have a solution in the compact case. Since then different results have confirmed the conjecture in various cases, some are quoted in the references of this paper. In particular the authors proved in [Int. Math. Res. Not. 2022, No. 12, 9499--9528 (2022; Zbl 1502.53080)] the conjecture under the condition that the manifold is a torus fibration over a 4-dimensional hypercomplex base. The paper under review is an extension of this result to spaces admitting general corank 4 hypercomplex fibrations. The methods use similar ideas, but are non-trivial generalizations of the ones in their previous paper.
Reviewer: Gueo Grantcharov (Miami)Rudin's theorem for model domainshttps://zbmath.org/1522.320402023-12-07T16:00:11.105023Z"Ourimi, Nabil"https://zbmath.org/authors/?q=ai:ourimi.nabilSummary: For any \(\mathcal{C}^\infty\)-smooth almost complex structure \(J^\prime\) on \(\mathbb{C}^{n+1}\), we prove that any proper holomorphic mapping from a model domain in \(\mathbb{C}^{n+1}\) to a bounded domain in \((\mathbb{C}^{n+1}, J^\prime)\), that has a \(\mathcal{C}^1\)-extension to the boundary is factored by a finite group of automorphisms \(\Gamma\), conjugate to a finite subgroup of \(Id_{\mathbb{C}}\times U(n)\). Further, we obtain rigidity results when additional conditions are imposed on the target domain. Our main result extends Rudin's theorem, on proper holomorphic mappings and finite reflection groups, to the almost complex case.Geometric similarity invariants of Cowen-Douglas operatorshttps://zbmath.org/1522.320472023-12-07T16:00:11.105023Z"Jiang, Chunlan"https://zbmath.org/authors/?q=ai:jiang.chunlan"Ji, Kui"https://zbmath.org/authors/?q=ai:ji.kui"Keshari, Dinesh Kumar"https://zbmath.org/authors/?q=ai:keshari.dinesh-kumarSummary: \textit{M. J. Cowen} and \textit{R.G. Douglas} [Acta Math. 141, 187--261 (1978; Zbl 0427.47016)] introduced a class of operators \(B_n (\Omega)\) (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in \(B_n (\mathbb{D})\) can be constructed from a given operator in \(B_1 (\mathbb{D})\). We also characterize a class of strongly irreducible weakly homogeneous operators in \(B_n (\mathbb{D})\).Elliptic coadjoint orbits of holomorphic typehttps://zbmath.org/1522.320532023-12-07T16:00:11.105023Z"Sekiguchi, Hideko"https://zbmath.org/authors/?q=ai:sekiguchi.hidekoSummary: This article proves that any elliptic coadjoint orbit of a semisimple Lie group carries a holomorphic bundle structure over a flag variety if the polarization is given by a \(\theta\)-stable parabolic subalgebra of holomorphic type. An application to the Penrose transform is given.Boundary points, minimal \(L^2\) integrals and concavity property \(V\): vector bundleshttps://zbmath.org/1522.320582023-12-07T16:00:11.105023Z"Guan, Qi'an"https://zbmath.org/authors/?q=ai:guan.qian"Mi, Zhitong"https://zbmath.org/authors/?q=ai:mi.zhitong"Yuan, Zheng"https://zbmath.org/authors/?q=ai:yuan.zhengSummary: In this article, for singular hermitian metrics on holomorphic vector bundles, we consider minimal \(L^2\) integrals on sublevel sets of plurisubharmonic functions on weakly pseudoconvex Kähler manifolds related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal \(L^2\) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a strong openness property of the modules and a twisted version, an effectiveness result of the strong openness property of the modules, and an optimal support function related to the modules.Primitive decomposition of Bott-Chern and Dolbeault harmonic \((k, k)\)-forms on compact almost Kähler manifoldshttps://zbmath.org/1522.320602023-12-07T16:00:11.105023Z"Holt, Tom"https://zbmath.org/authors/?q=ai:holt.tom"Piovani, Riccardo"https://zbmath.org/authors/?q=ai:piovani.riccardoSummary: We consider the primitive decomposition of \(\overline{\partial}\), \(\partial\), Bott-Chern and Aeppli-harmonic \((k, k)\)-forms on compact almost Kähler manifolds \((M,J,\omega)\). For any \(D \in \{ \overline{\partial}, \partial, \mathrm{BC}, \mathrm{A}\}\), it is known that the \(L^k P^{0,0}\) component of \(\psi\in\mathcal{H}^{k,k}_D\) is a constant multiple of \(\omega^k\) up to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the components \(L^{k-1} P^{1,1}\) and \(L^{k-2} P^{2,2}\) of the primitive decomposition. Focusing on dimension 8, we give a full description of the spaces \(\mathcal{H}^{2,2}_{\mathrm{BC}}\) and \(\mathcal{H}^{2,2}_{\mathrm{A}}\), from which follows \(\mathcal{H}^{2,2}_{\mathrm{BC}}\subseteq \mathcal{H}^{2,2}_{\partial}\) and \(\mathcal{H}^{2,2}_{\mathrm{A}}\subseteq \mathcal{H}^{2,2}_{\overline{\partial}}\). We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of a harmonic form \(\psi\in \mathcal{H}^{k,k}_D\) are not \(D\)-harmonic, showing that the primitive decomposition of \((k, k)\)-forms in general does not descend to harmonic forms.Harmonic morphisms and their Milnor fibrationshttps://zbmath.org/1522.320672023-12-07T16:00:11.105023Z"Ribeiro, M. F."https://zbmath.org/authors/?q=ai:ribeiro.maico-f"Araújo dos Santos, R. N."https://zbmath.org/authors/?q=ai:araujo-dos-santos.raimundo-nonato"Dreibelbis, D."https://zbmath.org/authors/?q=ai:dreibelbis.daniel"Griffin, M."https://zbmath.org/authors/?q=ai:griffin.mark|griffin.matt|griffin.malcolm-p|griffin.michael-j|griffin.maryclareSummary: In this paper, we study the relationships between harmonic morphisms and fibered structures in the local setting.CR-submanifolds of SQ-Sasakian manifoldhttps://zbmath.org/1522.320762023-12-07T16:00:11.105023Z"Yadav, Sarvesh Kumar"https://zbmath.org/authors/?q=ai:yadav.sarvesh-kumar"Hui, Shyamal Kumar"https://zbmath.org/authors/?q=ai:hui.shyamal-kumar"Iqbal, Mohd."https://zbmath.org/authors/?q=ai:iqbal.mohd-ashraf"Mandal, Pradip"https://zbmath.org/authors/?q=ai:mandal.pradip"Aslam, Mohd."https://zbmath.org/authors/?q=ai:aslam.mohdSummary: In this paper we discussed CR-submanifold of SQ-Sasakian manifold. Next, we considered Chaki pseudo parallel as well as Deszcz pseudo parallel CR-submanifold of SQ-Sasakian manifold. Further we studied almost Ricci soliton and almost Yamabe soliton with torse forming vector field on CR-subamnifold of SQ-Sasakian manifold using semi-symmetric metric connection.Real-analytic coordinates for smooth strictly pseudoconvex CR-structureshttps://zbmath.org/1522.320772023-12-07T16:00:11.105023Z"Kossovskiy, I."https://zbmath.org/authors/?q=ai:kossovskiy.ilya-g"Zaitsev, D."https://zbmath.org/authors/?q=ai:zaitsev.dmitriThis article studies the question of whether a strictly pseudoconvex smooth hypersurface \(M\) in some complex manifold \(X\) is diffeomorphic to a real analytic CR manifold. The authors refer to this property as being ``analytically regularizable'', since it makes a number of additional analytical tools available. The article presents examples that show that the question is non-trivial. The main result of the article is that a property called ``Condition E'' by the authors is equivalent to analytic regularizability. Condition E amounts to a condition on holomorphic extendability of a smooth function that can be explicitly described.
This starts from a local defining function for \(M\), which is formally complexified in order to define formal Segre varieties at a point \(p\in M\). Passing to jets, one obtains (local) embeddings of \(M\) into spaces of jets of hypersurfaces in \(X\). For the one-jets, the resulting embedding realizes \(M\) as a totally real submanifold and the embedding into \(2\)-jets can be viewed as being defined on this submanifold. Condition E says that this function is locally holomorphically extendable to the pseudoconvex side. In coordinates, this can be equivalently characterized as extendability of functions obtained from determinants of iterated partial derivatives of the defining function which are similar to the ones giving rise to the Monge-Ampère equation. Necessity and sufficiency of Condition E are proved separately, via an associated system of differential equations, respectively via a careful study of a certain foliation of the jet bundle.
Reviewer: Andreas Cap (Wien)\(C^{1,1}\)-rectifiability and Heintze-Karcher inequality on \(\mathbf{S}^{n+1}\)https://zbmath.org/1522.351332023-12-07T16:00:11.105023Z"Zhang, Xuwen"https://zbmath.org/authors/?q=ai:zhang.xuwenSummary: In this paper, by isometrically embedding \((\mathbf{S}^{n+1}, g_{\mathbf{S}^{n+1}})\) into \(\mathbf{R}^{n+2}\), and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are \(C^{1,1}\)-rectifiable. As a by-product, we establish a Heintze-Karcher-type inequality on sphere.On the Alesker-Verbitsky conjecture on hyperKähler manifoldshttps://zbmath.org/1522.355292023-12-07T16:00:11.105023Z"Dinew, Sławomir"https://zbmath.org/authors/?q=ai:dinew.slawomir"Sroka, Marcin"https://zbmath.org/authors/?q=ai:sroka.marcinSummary: We solve the quaternionic Monge-Ampère equation on hyperKähler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperKähler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi-Yau type theorem for HKT metrics is discussed.Classification of solutions for mixed order conformally system with Hartree-type nonlinearity in \(\mathbb{R}^n\)https://zbmath.org/1522.355542023-12-07T16:00:11.105023Z"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Peng, Shaolong"https://zbmath.org/authors/?q=ai:peng.shaolongSummary: In this paper, we consider the following mixed order conformally invariant system with Hartree-type nonlinearity:
\[\begin{cases}
(-\Delta)^su(x)=\left(\frac{1}{|x|^\sigma}*v^{\frac{2n-\sigma}{n-2}}\right)v^{\frac{n+2s-\sigma}{n-2}}(x), & \text{in }\mathbb{R}^n, \\
(-\Delta)v(x)=u^{\frac{n+2}{n-2s}}(x), & \text{in }\mathbb{R}^n,
\\ u\ge 0,\ v\ge 0, & \text{in }\mathbb{R}^n,
\end{cases}\tag{0.1}\]
where \(0<s=:m+\frac{\alpha}{2}<+\infty,m\) is a integer, \(0<\alpha\le 2\), \(n\ge 3\), \(\sigma\in(0,n)\). We first prove the equivalence of the PDEs system and the IEs system. Then we give the classification of the nonnegative solutions to the system (0.1) by using the method of moving spheres. Finally, we prove Liouville-type theorems results for system (0.1) in the critical and supercritical-order cases (i.e. \(\frac{n}{2}<s=:m+\frac{\alpha}{2}<+\infty)\), respectively.Sampling the X-ray transform on simple surfaceshttps://zbmath.org/1522.355932023-12-07T16:00:11.105023Z"Monard, François"https://zbmath.org/authors/?q=ai:monard.francois"Stefanov, Plamen"https://zbmath.org/authors/?q=ai:stefanov.plamen-dThe paper studies the problem of proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces, and illustrates the theory on simple geodesic disks of constant curvature. The notion of bandlimited and semiclassical bandlimited functions is introduced, followed by a discussion of a general strategy to avoid aliasing and derive minimal sampling rates by using the Nyquist-Shannon sampling theorem and its semiclassical version.
The authors proceed to lay out the theory of a general simple surface, in particular, they look for convenient representations of the canonical relation of the X-ray transform in two coordinate systems. The first one is the fan-beam coordinate system, and the second is generalized parallel coordinates in the Euclidean case. They quantify the quality of a sampling scheme depending on geometric parameters of the surface and the coordinate system used to represent the space of geodesics.
For the purpose of numerical illustration, the authors focus on simple geodesic disks of constant curvature, or constant curvature disks for short, a two-parameter family of simple surfaces indexed by their diameter and curvature. The analysis is based on the asymptotic sampling theory, which was developed by the second author in a previous paper, and uses semiclassical microlocal methods as as the sampling step size gets smaller and smaller.
Reviewer: Ahmed I. Zayed (Chicago)Global Lipschitz stability for inverse problems of wave equations on Lorentzian manifoldshttps://zbmath.org/1522.355972023-12-07T16:00:11.105023Z"Takase, Hiroshi"https://zbmath.org/authors/?q=ai:takase.hiroshiSummary: We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The method used in this paper is widely known as the Bukhgeim-Klibanov method. However, the conventional method is not sufficient for the application to the hyperbolic partial differential equation with time-dependent coefficients to obtain the Lipschitz stability, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability.Invariant volume forms of geodesic, potential, and dissipative systems on a tangent bundle of a four-dimensional manifoldhttps://zbmath.org/1522.370392023-12-07T16:00:11.105023Z"Shamolin, M. V."https://zbmath.org/authors/?q=ai:shamolin.m-vSummary: Complete sets of invariant differential forms of phase volume for homogeneous dynamical systems on tangent bundles of smooth four-dimensional manifolds are presented. The connection between the existence of these invariants and the complete set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The introduced force fields make the considered systems dissipative with dissipation of different signs and generalize previously considered fields.Retraction note to: ``On the dynamics of hypersurfaces foliated by \(n-2\)-dimensional oriented hyperspheres in \(\mathbb{S}^n\)''https://zbmath.org/1522.370442023-12-07T16:00:11.105023ZFrom the text: Following an investigation by the Taylor \& Francis Research Integrity and Ethics team, it was discovered post-publication that the article [\textit{L. Y. Fan} and \textit{M. A. López}, ibid. 25, No. 9--10, 1218--1238 (2019; Zbl 1435.37048)] was not peer reviewed appropriately, in line with the Journal's peer review standards and policy. As the stringency of the peer review process is core to the integrity of the publication process, the Editor and Publisher have taken the decision to retract this article. The handling Guest Editor was unable to confirm how reviewers were invited and whether the handling of this paper was in line with the expected standards of the journal. The journal has not received a response from the reviewers.Polygon recutting as a cluster integrable systemhttps://zbmath.org/1522.370762023-12-07T16:00:11.105023Z"Izosimov, Anton"https://zbmath.org/authors/?q=ai:izosimov.antonSummary: Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with opposite orientation. Recuttings along different diagonals generate an action of the affine symmetric group on the space of polygons. We show that this action is given by cluster transformations and is completely integrable. The integrability proof is based on interpretation of recutting as refactorization of quaternionic polynomials.Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchyhttps://zbmath.org/1522.370802023-12-07T16:00:11.105023Z"Takasaki, Kanehisa"https://zbmath.org/authors/?q=ai:takasaki.kanehisaThe author proposes a generalization of the integrable hierarchy of the intermediate long wave equation (ILW). This hierarchy can be viewed as a reduction of the lattice Kadomtsev-Petviashvili (KP) hierarchy and depends on a certain parameter. By setting that parameter to zero, the author proves that the generalized ILW hierarchy leads to an extended lattice Gelfand-Dickey hierarchy. In a similar manner it can be proved that the equivariant one-dimensional (resp., bigraded) Toda hierarchy reduces to the extended one-dimensional (resp., bigraded) Toda hierarchy.
Another important result in the paper concerns the integration of the considered hierarchy. Starting from the soliton solutions of the lattice KP hierarchy, the author obtains certain conditions that lead to soliton solutions of the generalized ILW hierarchy. An alternative approach of integration is based upon a factorization problem of difference solution-generating operators related to the lattice KP hierarchy with some reduction imposed. The described procedure allows one to construct all the solutions to the generalized ILW hierarchy. A special attention is paid to the case when the solution-generating operators are analogous to those appearing in the fermionic description of the equivariant Gromov-Witten theory of \(\mathbb{C}P^1\).
Reviewer: Tihomir Valchev (Sofia)Symplectic groupoids for Poisson integratorshttps://zbmath.org/1522.370852023-12-07T16:00:11.105023Z"Cosserat, Oscar"https://zbmath.org/authors/?q=ai:cosserat.oscarSummary: We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.Widths and entropy of sets of smooth functions on compact homogeneous manifoldshttps://zbmath.org/1522.410122023-12-07T16:00:11.105023Z"Kushpel, Alexander"https://zbmath.org/authors/?q=ai:kushpel.alexander-k"Taş, Kenan"https://zbmath.org/authors/?q=ai:tas.kenan"Levesley, Jeremy"https://zbmath.org/authors/?q=ai:levesley.jeremyA general method is developed to calculate entropy and \(n\)-widths of sets of smooth functions on an arbitrary compact homogeneous Riemannian manifold \(\mathbb{M}^d\), which is based on study of geometric characteristics of norms induced by subspaces of harmonics on \(\mathbb{M}^d\). As an application, the orders of entropy and \(n\)-widths of Sobolev's classes \(W_p^\gamma(\mathbb{M}^d)\) and their generalisations in \(L_q(\mathbb{M}^d)\) are obtained; the results are sharp for \(1<p,q<\infty\), while for \(p,q=1,\infty\) the sharpness is up to a logarithmic factor.
Reviewer: Andriy Prymak (Winnipeg)Extremal curves in the conformal space and in an associated bundlehttps://zbmath.org/1522.490382023-12-07T16:00:11.105023Z"Krivonosov, Leonid N."https://zbmath.org/authors/?q=ai:krivonosov.leonid-nikolaevich"Luk'yanov, Vyacheslav A."https://zbmath.org/authors/?q=ai:lukyanov.vyacheslav-anatolevich"Voloskova, Lubov V."https://zbmath.org/authors/?q=ai:voloskova.lubov-vSummary: Time-like and null extremal curves are computed in the conformal space for bundles with fibers \(P_5\) and \(P_5\wedge P_5\).On the entropy of Hilbert geometries of low regularitieshttps://zbmath.org/1522.510112023-12-07T16:00:11.105023Z"Cristina, Jan"https://zbmath.org/authors/?q=ai:cristina.jan"Merlin, Louis"https://zbmath.org/authors/?q=ai:merlin.louisThe aim of this paper is to prove two main results on the volume entropy of Hilbert geometries. The first one states that if \(\Omega\) is a convex and relatively compact domain of \({\mathbb R}^2\) which is Ahlfors \(\alpha\)-regular, then \(h(\Omega) = \frac{2\alpha}{\alpha +1}\), where \(h(\Omega)\) stands for the volume growth entropy.
The second results strengthens the first main theorem of \textit{G. Berck} et al. [Pac. J. Math. 245, No. 2, 201--225 (2010; Zbl 1204.52003)] by weakening the assumption of \(C^{1,1}\)-regularity of the boundary of the convex set \(\Omega\).
Reviewer: Victor V. Pambuccian (Glendale)Geometry of the ovoids: reptilian eggs and similar symmetric formshttps://zbmath.org/1522.510142023-12-07T16:00:11.105023Z"Mladenova, Clementina D."https://zbmath.org/authors/?q=ai:mladenova.clementina-d"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mIn analogy to the conic sections studied by Apollonius of Perga, the second century AD Greek geometer Perseus studied spiric sections, intersections of a torus with a plane that is parallel to its rotational symmetry axis. These were studied analytically by the second author in [J. Geom. Symmetry Phys. 58, 81--97 (2020; Zbl 1471.53004)], where their use as a geometrical model of the egg has been emphasized. Here, the authors derive explicit formulas for the volume, surface area and the curvatures of the egg-mimicking shapes (for avian or reptilian eggs, depending on the values of some parameters) and then compare them with experimental data.
For the entire collection see [Zbl 1516.53003].
Reviewer: Victor V. Pambuccian (Glendale)Yet another mathematical model of eggs: two-parametric Brandt's shapeshttps://zbmath.org/1522.510152023-12-07T16:00:11.105023Z"Mladenova, Clementina D."https://zbmath.org/authors/?q=ai:mladenova.clementina-d"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mThis is part of a ``series of short reviews in which the existing ``models'' [for eggs] are covered in some depth and where possible -- appropriately extended'' that the author have recently embarked upon (``an almost exhaustive list of such ``models'' can be found in \textit{W. Hortsch} [Alte und neue Eiformeln in der Geschichte der Mathematik. München: Selbstverlag Hortsch (1990)]''). The model analysed here was proposed in an obscure venue by \textit{G. Brandt} [The research of an equation of a shell formed by the two-focus curve, in: Sb. tr. VZPI: ``Stroitelstvo i arhitektura''. Moscow: VZBI. 76--86 (1973)]. It is a surface of revolution described by
\[
z^2 + y^2 = \frac{3x(2a - x)((x+a)^2-c^2)}{4(x+a)^2}\quad x\in[0, 2a]
\]
in which \(a > \)c are real positive parameters
Reviewer: Victor V. Pambuccian (Glendale)Integral geometry of pairs of hyperplanes or lineshttps://zbmath.org/1522.520122023-12-07T16:00:11.105023Z"Hug, Daniel"https://zbmath.org/authors/?q=ai:hug.daniel"Schneider, Rolf"https://zbmath.org/authors/?q=ai:schneider.rolf-gFrom the abstract: ``Crofton's formula of integral geometry evaluates the total motion invariant measure of the set of \(k\)-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body.''
Recently, questions, similar to Crofton's, were asked about intersections of a convex domain with pairs of lines [\textit{J. Cufí} et al., Rend. Circ. Mat. Palermo (2) 69, No. 3, 1115--1130 (2020; Zbl 1459.52005)].
This article extends this topic to higher dimensions. The authors prove that if we intersect two convex bodies of constant width with pairs of hyperplanes then a certain integral over a motion-invariant measure only depends on the widths of the bodies.
Similar result is proven for bodies of constant brightness and intersections with pairs of lines. Some converse results are also proven. Formulae for general convex bodies are also obtained.
Reviewer: Nikita Kalinin (Guangdong)Prescribing the Gauss curvature of convex bodies in hyperbolic spacehttps://zbmath.org/1522.520182023-12-07T16:00:11.105023Z"Bertrand, Jérôme"https://zbmath.org/authors/?q=ai:bertrand.jerome"Castillon, Philippe"https://zbmath.org/authors/?q=ai:castillon.philippeThe hyperbolic version of A. D. Alexandrov's problem is solved: A finite measure \(\mu\) on the sphere \(S^m\) is the curvature measure of some convex body of the hyperbolic space iff three conditions are satisfied. One of them is the Alexandrov's condition (coming from the Euclidean A. D. Alexandrov's problem), the other two conditions are specific for the hyperbolic space. Under these conditions, the body is unique.
Reviewer: Gaiane Panina (Sankt-Peterburg)Manifolds, vector fields, and differential forms. An introduction to differential geometryhttps://zbmath.org/1522.530012023-12-07T16:00:11.105023Z"Gross, Gal"https://zbmath.org/authors/?q=ai:gross.gal"Meinrenken, Eckhard"https://zbmath.org/authors/?q=ai:meinrenken.eckhardThis book is intended to be a modern introduction to the basics of differential geometry, accessible to undergraduate and master students. From my point of view, this goal is achieved, the book being very well structured and supported by illustrative examples and problems. It is divided into 9 chapters and 3 appendices.
The first chapter is introductory, containing some historical landmarks, the presentation of the concept of manifolds, aspects regarding manifolds in Euclidean spaces, as well as intrinsic descriptions. The second chapter presents definitions and fundamental properties concerning the geometry of manifolds. The notions of atlases and charts are discussed first, then the definition of manifolds and numerous examples are presented. After a review of compactness and orientability of manifolds, the classical procedures to build new examples of manifolds are presented. Chapter 3 deals with smooth maps between manifolds. After discussing functions on manifolds, a simple criterion for the Hausdorff condition is provided using smooth functions. Then smooth maps are presented and several examples are given. This chapter ends with a complete description of the Hopf fibration. Chapter 4 deals with submanifolds. Special attention is given to the study of local diffeomorphisms, submersions, quotient maps and immersions. Tangent spaces are the topic of Chapter 5. This chapter starts with the intrinsic definition of tangent spaces and continues with presenting basic properties of tangent maps. Finally, tangent spaces of submanifolds are investigated and an example is provided. Chapter 6 discusses vector fields as derivations, Lie brackets, related vector fields and flows of vector fields. At the end of this chapter, the Frobenius theorem is proved. Chapter 7 addresses differential forms, while the next chapter deals with the integration of differential forms. Stokes theorem, winding numbers, mapping degrees and volume forms are investigated in Chapter 8. The last chapter is devoted to the study of vector bundles and their sections. The three appendices at the end of the book present fundamental results from set theory, algebra as well as topological properties of manifolds. The volume ends with a bibliography, including some classics in the field of differential geometry.
In the reviewer's opinion, this book will be of great interest for undergraduate students, master students, and also helpful for instructors.
Reviewer: Gabriel Eduard Vilcu (București)Möbius inversion of surfaces in the Minkowski 3-spacehttps://zbmath.org/1522.530022023-12-07T16:00:11.105023Z"do Couto Fernandes, Marco Antônio"https://zbmath.org/authors/?q=ai:do-couto-fernandes.marco-antonioAuthor's abstract: We define and present some properties of the Möbius inversion of surfaces in the Minkowski \(3\)-space. We prove that the Möbius inversion preserves the lines of principal curvature and the locus of points where the metric is degenerate, but it does not preserve the parabolic set. For ovaloids, we show that it is possible to translate the surface so that the inversion remains an ovaloid.
Reviewer: Friedrich Manhart (Wien)Measurement and calculation on conformable surfaceshttps://zbmath.org/1522.530032023-12-07T16:00:11.105023Z"Has, Aykut"https://zbmath.org/authors/?q=ai:has.aykut"Yılmaz, Beyhan"https://zbmath.org/authors/?q=ai:yilmaz.beyhanSummary: In this study, some basic concepts related to the surface are examined with the help of conformable fractional analysis. As known, the best thing that makes fractional analysis popular is that it gives numerically more approximate results compared to classical analysis. For this reason, the concepts that enable us to make calculations based on the measurement on the surface have been redefined to give more numerical results with conformable fractional analysis. In addition, with the help of fractional analysis, it is explained which concepts are changed or not. Finally, an example is given to better understand the obtained results and its graph is drawn with the help of Mathematica. The reason for using conformable fractional analysis in this study is that it is more suitable for the algebraic structure of differential geometry.Hypersurfaces satisfying \(\triangle \overrightarrow{H} = \lambda \overrightarrow{H}\) in \(\mathbb{E}_s^5\)https://zbmath.org/1522.530042023-12-07T16:00:11.105023Z"Gupta, Ram Shankar"https://zbmath.org/authors/?q=ai:gupta.ram-shankar"Arvanitoyeorgos, Andreas"https://zbmath.org/authors/?q=ai:arvanitoyeorgos.andreasThe paper considers a class of hypersurfaces \(M^4_r\) in the pseudo-Euclidean space \(\mathbb{R}^5_s\), where \(0 \leq s \leq 5\) and \(0 \leq r \leq 4\) stand for the indices of the metric \(g_E\) on \(\mathbb{R}^5_s\) and the metric \(g_M\) on \(M^4_r\) induced by \(g_E\), respectively. The class consists of the hypersurfaces \(M^4_r \subset \mathbb{R}^5_s\) satisfying
\[
\Delta \vec{H} = \lambda \vec{H},\tag{1}
\]
where \(\Delta\) is the Laplace operator with respect to the metric \(g_M\), \(\vec{H}\) is the mean curvature vector of \(M^4_r\) and \(\lambda\) is a real constant. As a particular case, a hypersurface satisfying (1) with \(\lambda=0\) is called biharmonic. In this context, the paper presents three main results.
The first result states that a hypersurface \(M^4_r \subset \mathbb{R}^5_s\), whose mean curvature vector satisfies (1), with diagonal shape operator and distinct principal curvatures, has constant mean curvature, constant norm of the second fundamental form and constant scalar curvature. As a consequence of this theorem and the main result of \textit{J. Liu} and \textit{C. Yang} [J. Math. Anal. Appl. 419, No. 1, 562--573 (2014; Zbl 1296.53126)], the authors conclude that a hypersurface \(M^4_r \subset \mathbb{R}^5_s\) satisfying (1) and with diagonal shape operator, has constant mean curvature, constant norm of the second fundamental form and constant scalar curvature. This is the content of the second main result.
Finally, the third main result asserts that a biharmonic hypersurface \(M^4_r \subset \mathbb{R}^5_s\), with diagonal shape operator, must be minimal. This result is related to a conjecture due to \textit{B.-Y. Chen} [Soochow J. Math. 17, No. 2, 169--188 (1991; Zbl 0749.53037)], which states that the only biharmonic submanifolds of Euclidean spaces are the minimal submanifolds.
Reviewer: João Paulo dos Santos (Brasília)Minimal surfaces in the three-sphere by desingularizing intersecting Clifford torihttps://zbmath.org/1522.530052023-12-07T16:00:11.105023Z"Kapouleas, Nikolaos"https://zbmath.org/authors/?q=ai:kapouleas.nikolaos"Wiygul, David"https://zbmath.org/authors/?q=ai:wiygul.davidFrom the authors summary: ``For each integer \(k\ge 2\) we apply PDE gluing methods to desingularize certain collections of intersecting Clifford tori, thus producing sequences of minimal surfaces embedded in the round three-sphere. The collections of the Clifford tori we use consist of either \(k\) Clifford tori intersecting with maximal symmetry along two orthogonal great circles (lying on orthogonally complementary two-planes) or of the same \(k\) Clifford tori supplemented by an additional Clifford torus equidistant from the original two circles of intersection so that the latter torus orthogonally intersects each of the former \(k\) tori along a pair of disjoint orthogonal circles. The former two circles get desingularized by using singly periodic Karcher-Scherk towers of order \(k\) as models, so that after rescaling the sequences of minimal surfaces converge smoothly on compact subsets to the Karcher-Scherk tower of order \(k\). Near the other \(2k\) circles (in the latter case) the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type, where the number of handles desingularizing each circle is the same, resemble surfaces constructed by \textit{J. Choe} and \textit{M. Soret} [ibid. 364, No. 3--4, 763--776 (2016; Zbl 1341.49052))] by different methods. There are many new examples which are more complicated and on which the numbers of handles for the two circles differ. All examples of the latter type are new as well.'' It would be interesting to know what the DPW representation of these surfaces are.
Reviewer: Ivan C. Sterling (St. Mary's City)Correction to: ``Equilibrium of surfaces in a vertical force field''https://zbmath.org/1522.530062023-12-07T16:00:11.105023Z"Martínez, Antonio"https://zbmath.org/authors/?q=ai:martinez.antonio-b|martinez.antonio-olivas"Martínez-Triviño, A. L."https://zbmath.org/authors/?q=ai:martinez-trivino.antonio-luisFrom the text: In our previous paper [ibid. 19, No. 1, Paper No. 3, 28 p. (2022; Zbl 1500.53012), Section 3.2], we describe the family of titled \([\varphi, \vec{e}_3]\)-catenary cylinders as surfaces obtained from a \([\varphi, \vec{e}_3]\)-catenary cylinder \(\Sigma\), by rotation of angle \(\theta \in]0, \pi/2[\) about the \(x\)-axis and dilation by scale factor \(1/\cos\theta\). The authors state that the resulting surface, \(\widetilde{\Sigma}\), is always a \([\varphi, \vec{e}_3]\)-minimal surface, but this is only true when the starting \([\varphi, \vec{e}_3]\)-catenary cylinder is a grim reaper translating soliton, which follows directly from the relationship between their mean curvatures.Simons-type inequalities for minimal surfaces with constant Kähler angle in a complex hyperquadrichttps://zbmath.org/1522.530072023-12-07T16:00:11.105023Z"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.12"Fei, Jie"https://zbmath.org/authors/?q=ai:fei.jie"Jiao, Xiaoxiang"https://zbmath.org/authors/?q=ai:jiao.xiaoxiangA complex submanifold of \(\mathbb{CP}^{n+1}\), \[\mathbf{Q}_n=\{[Z=(z_1,\dots,z_{n+2}]\in \mathbb{CP}^{n+1}|z_1^2+\ldots+z_{n+2}^2=0\},\] where \(Z=(z_1,\dots,z_{n+2})\) are the homogeneous coordinates of \(\mathbb{CP}^{n+1}\), is called a complex hyperquadric.
In this paper, the authors consider compact minimal surfaces \(M\) in \(\mathbf{Q}_n\) with constant Kähler angle and provide many Simons-type inequalities for \(M\). Here is one theorem in the paper.
Theorem. Let \(M\) be a compact surface without boundary and \(f:M\rightarrow \mathbf{Q}_{n}\) a minimal immersion (non \(\pm\)holomorphic) with constant Kähler angle \(\theta\). Then
\[
\int_M [(K-1)A+9\cos^2\theta-4\cos\theta(\tau_X^2-\tau_Y^2)]*1\geq 0,
\]
where
\[
A = 3K-2-13\cos^2\theta+4\tau_X^2+4\tau_Y^2-16\tau_{XY}^2.
\]
They also consider totally real minimal immersions from \(S^2\) to \(\mathbf{Q}_n\) and holomorphic immersions. At the end, as an application, they determine all totally real minimal two-spheres with a special inequality on the second fundamental form.
Reviewer: Yun Myung Oh (Berrien Springs)Computational and qualitative aspects of evolution of curves driven by curvature and external forcehttps://zbmath.org/1522.530082023-12-07T16:00:11.105023Z"Mikula, Karol"https://zbmath.org/authors/?q=ai:mikula.karol"Ševčovič, Daniel"https://zbmath.org/authors/?q=ai:sevcovic.danielSummary: We propose a direct method for solving the evolution of plane curves satisfying the geometric equation \(v=\beta(x,k,\nu)\) where \(v\) is the normal velocity, \(k\) and \(\nu\) are the curvature and tangential angle of a plane curve \(\Gamma\subset\mathbb{R}^2\) at a point \(x\in\Gamma\). We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves. The governing equations include a nontrivial tangential velocity functional yielding uniform redistribution of grid points along the evolving family of curves preventing thus numerically computed solutions from forming various instabilities. We also propose a full space-time discretization of the governing system of equations and study its experimental order of convergence. Several computational examples of evolution of plane curves driven by curvature and external force as well as the geodesic curvatures driven evolution of curves on various complex surfaces are presented in this paper.Lower bounds for the length of the second fundamental form via the first eigenvalue of the \(p\)-Laplacianhttps://zbmath.org/1522.530092023-12-07T16:00:11.105023Z"dos Santos, Fábio R."https://zbmath.org/authors/?q=ai:dos-santos.fabio-reis"Soares, Matheus N."https://zbmath.org/authors/?q=ai:soares.matheus-nLet \((M^n,g)\) be an \(n\)-dimensional closed minimal submanifold in \(\mathbb S^m\), the unit \(m\)-sphere, and \(u\) be an eigenfunction of the \(p\)-Laplacian operator on \(M^n\) associated to the eigenvalue \(\lambda\). The authors prove that
\[
\int_{M^n}(S+\alpha_{n,p}\lambda^{p/2}-n)|\nabla u|^{2(p-1)}d\mu_g\ge 0,
\]
where \(p>2\), \(\alpha_{n,p}\) is an explicit constant relying on \(n,p\), and \(S\) represents the squared norm of the second fundamental form. Also, they prove an analogous result when \(M^n\) is a compact minimal submanifold in \(\mathbb S^m\).
Reviewer: Mohammed El Aïdi (Bogotá)Projective invariants of imageshttps://zbmath.org/1522.530102023-12-07T16:00:11.105023Z"Olver, Peter J."https://zbmath.org/authors/?q=ai:olver.peter-jSummary: The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order \(\leq 3\), it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.Approximation of the Willmore energy by a discrete geometry modelhttps://zbmath.org/1522.530112023-12-07T16:00:11.105023Z"Gladbach, Peter"https://zbmath.org/authors/?q=ai:gladbach.peter"Olbermann, Heiner"https://zbmath.org/authors/?q=ai:olbermann.heinerThe Willmore energy is an energy attached to an embedding of a surface in \(\mathbb{R}^3\). Aside from giving a notion of how much a surface deviates from a round sphere, the concept of Willmore energy has applications in physics and engineering, as it shows up as a limit of the bending energy in the Seung-Nelson model associated to an (immersed) triangulated surface in \(\mathbb{R}^3\).
One of the main motivations for discrete differential geometry is to find discrete analogues of notions from the smooth setting. There are many ways to discretize an object of the smooth world and one of the main goals is to find a discretization that preserves some of the structural properties of the object. A side effect of this is that good discretization algorithms are often relatively hard to find. The main goal of the paper is to investigate the convergence of the energies defined on discrete triangulated surfaces. In the literature, different energies associated with immersions of triangulated surfaces that are mutually asymptotically equivalent have been proposed and investigated, and conditions that imply their convergence to the Willmore energy have been found.
The current paper defines a new energy, associated to a triangulated surface, that is equivalent to previous definitions in the sense that the difference between them vanishes in the limit of ever smaller triangles (if the limiting surface is sufficiently smooth). The main result of the paper is that this energy converges in a precise sense to the Willmore energy if certain regularity assumptions are satisfied. The authors consider a particular topology for their convergence, and their convergence results correspond to \(\Gamma\)-convergence. The results of the paper are more general than previous convergence results on related energies, eliminating many of the more stringent assumptions. The main two remaining regularity assumptions for the convergence results are the Delaunay property, whereby the smallest ball containing the vertices of a triangle of the surface does not contain any other vertices of the triangulation, and some control over the minimum angles that show up in the triangulation.
There are two main sets of results that can be roughly summarized as follows:
(1) For a sufficiently well-behaved sequence of triangulated surfaces, the Willmore energy of the smooth limit surface is smaller than or equal to the limit of the energies of the approximating surfaces;
(2) For a given smooth surface, there is a sequence of well-behaved approximating surfaces such that the limit of the discrete energies of these surfaces is equal to the Willmore energy.
The paper concludes with an example illustrating the necessity of the Delaunay condition for the results of the paper.
Reviewer: Benedikt Kolbe (Bonn)Alpha geodesic distances for clustering of shapeshttps://zbmath.org/1522.530122023-12-07T16:00:11.105023Z"De Sanctis, Angela A."https://zbmath.org/authors/?q=ai:de-sanctis.angela-a"Gattone, Stefano A."https://zbmath.org/authors/?q=ai:gattone.stefano-antonio"Oikonomou, Fotios D."https://zbmath.org/authors/?q=ai:oikonomou.fotios-dSummary: According to Information Geometry, we represent landmarks of a complex shape, as probability densities in a statistical manifold where geometric structures from \(\alpha \)-connections are considered. In particular the 0-connection is the Riemannian connection with respect to the Fisher metric. In the setting of shapes clustering, we compare the discriminative power of different shapes distances induced by geodesic distances derived from \(\alpha \)-connections. The methodology is analyzed in an application to a data set of aeroplane shapes.Colesanti type inequalities for affine connectionshttps://zbmath.org/1522.530132023-12-07T16:00:11.105023Z"Huang, Guangyue"https://zbmath.org/authors/?q=ai:huang.guangyue"Ma, Bingqing"https://zbmath.org/authors/?q=ai:ma.bingqing"Zhu, Mingfang"https://zbmath.org/authors/?q=ai:zhu.mingfangThe authors consider a generalisation, for a 2-parameter family of affine connections, of the classical Reilly type integral formula, which was originally established for Riemannian connections, and apply the generalized formula to a Colesanti type inequality.
Namely, they consider a triple \((M, \bar g, e^u) \,,\) where \((M, \bar g)\) is a smooth Riemannian manifold and \(u\) a smooth function of \(M \,.\) Then, by means of two real constants \(\alpha\) and \(\gamma \,,\) they define a 2-parameter family of affine connections \[
D^{\alpha,\gamma}_X Y = \bar\nabla_X Y + ad u(X) Y + ad u(Y) X + \gamma \bar g (X,Y) \bar\nabla u.
\]
In this framework, under certain conditions, they prove three inequalities holding for any smooth function \(f\) on the boundary \(\partial M\) and three inequalities holding for the first nonzero eigenvalue of the Laplacian.
Indeed, for particular values of the parameters, these formulas give classical formulas.
Reviewer: Marco Modugno (Firenze)Contact instantons and partial connectionshttps://zbmath.org/1522.530142023-12-07T16:00:11.105023Z"Udomlertsakul, Nathapon"https://zbmath.org/authors/?q=ai:udomlertsakul.nathapon"Wang, Shuguang"https://zbmath.org/authors/?q=ai:wang.shuguang|wang.shuguang.1Summary: We study instanton equations on contact manifolds from the point of view of partial connections. Results of \textit{H. Urakawa} [Math. Z. 216, No. 4, 541--573 (1994; Zbl 0815.32008)] and \textit{S. Dragomir} and \textit{H. Urakawa} [Interdiscip. Inf. Sci. 6, No. 1, 41--52 (2000; Zbl 0959.58020)] are generalized from strongly pseudo-convex CR manifolds to contact manifolds.Extremally Ricci pinched \(G_2\)-structures on Lie groupshttps://zbmath.org/1522.530152023-12-07T16:00:11.105023Z"Lauret, Jorge"https://zbmath.org/authors/?q=ai:lauret.jorge"Nicolini, Marina"https://zbmath.org/authors/?q=ai:nicolini.marinaA closed \(G_2\)-structure on a seven-dimensional manifold \(M\) is defined by a positive closed three-form \(\varphi\) on \(M\). If the torsion of a closed \(G_2\)-structure \(\varphi\) given by
\[
\tau = -{*}d * \varphi
\]
satisfies the neat equation
\[
d\tau =\frac{1}{6}(|\tau |^2\varphi + *(\tau \wedge \tau )),
\]
then the manifold is called extremally Ricci pinched (abbreviated as ERP).
Only two examples of extremally Ricci pinched \(G_2\)-structures could be found in the published literature and they were both homogeneous.
The authors study in this paper the existence of ERP \(G_2\)-structures on Lie groups and give structural conditions on the Lie algebra. They obtain three new examples that are all steady Laplacian solitons. They also study the deformation and rigidity of these structures.
These examples are complete and exhaustive thanks to the classification result obtained by the authors in [Ann. Mat. Pura Appl. (4) 199, No. 6, 2489--2510 (2020; Zbl 1451.53066)].
One may wonder why this article does not acknowledge the classification paper [loc. cit.], as both articles are similar in their findings. By comparing the dates of these papers, the result of the classification would have been known before the acceptance of this paper and therefore it could have incorporated the result of the classification, thus reducing duplication of material.
Reviewer: Matthew Randall (Nanjing)Some types of slant submanifolds of bronze Riemannian manifoldshttps://zbmath.org/1522.530162023-12-07T16:00:11.105023Z"Acet, Bilal Eftal"https://zbmath.org/authors/?q=ai:acet.bilal-eftal"Acet, Tuba"https://zbmath.org/authors/?q=ai:acet.tubaSummary: The aim of this article is to examine some types of slant submanifolds of bronze Riemannian manifolds. We introduce hemi-slant submanifolds of a bronze Riemannian manifold. We obtain integrability conditions for the distribution involved in quasi hemi-slant submanifold of a bronze Riemannian manifold. Also, we give some examples about this type submanifolds.Properties of warped product gradient Yamabe solitonshttps://zbmath.org/1522.530172023-12-07T16:00:11.105023Z"Tokura, Willian"https://zbmath.org/authors/?q=ai:tokura.willian-isao|tokura.willian"Barboza, Marcelo"https://zbmath.org/authors/?q=ai:barboza.marcelo-bezerra"Adriano, Levi"https://zbmath.org/authors/?q=ai:adriano.levi-rosa"Pina, Romildo"https://zbmath.org/authors/?q=ai:pina.romildo-da-silvaSummary: In this paper, we study gradient Yamabe solitons realized as warped product manifolds. We apply the maximum principle to find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product manifold. By slightly modifying Li-Yau's technique, we can handle the drifted Laplacian and thus find different gradient estimates for the warping function according to the sign of the (constant) scalar curvature of the fiber manifold. We close the article with a theorem stating the nonexistence of gradient Yamabe solitons on top of warped products with a base manifold satisfying certain analytical hypothesis.Front asymptotics of a flat sub-Riemannian structure on the Engel distributionhttps://zbmath.org/1522.530182023-12-07T16:00:11.105023Z"Bogaevsky, I. A."https://zbmath.org/authors/?q=ai:bogaevskij.ilya-aleksandrovichSummary: We approximate the front of a flat sub-Riemannian structure on the Engel distribution in a neighborhood of a non-subanalytic singularity by the front of a control system integrable in elementary functions. As a corollary, we find the asymptotics of the exponential map of a flat sub-Riemannian structure on the Engel distribution.Abnormal trajectories in the sub-Riemannian \((2,3,5,8)\) problemhttps://zbmath.org/1522.530192023-12-07T16:00:11.105023Z"Sachkov, Yu. L."https://zbmath.org/authors/?q=ai:sachkov.yuri-l"Sachkova, E. F."https://zbmath.org/authors/?q=ai:sachkova.elena-fSummary: Abnormal trajectories are of particular interest for sub-Riemannian geometry, because the most complicated singularities of the sub-Riemannian metric are located just near such trajectories. Important open questions in sub-Riemannian geometry are to establish whether the abnormal length minimizers are smooth and to describe the set filled with abnormal trajectories starting from a fixed point. For example, the Sard conjecture in sub-Riemannian geometry states that this set has measure zero. In this paper, we consider this and other related properties of such a set for the left-invariant sub-Riemannian problem with growth vector \((2,3,5,8)\). We also study the global and local optimality of abnormal trajectories and obtain their explicit parametrization.Positive mass theorem with arbitrary ends and its applicationhttps://zbmath.org/1522.530202023-12-07T16:00:11.105023Z"Zhu, Jintian"https://zbmath.org/authors/?q=ai:zhu.jintianThe author proves a generalization of the positive mass theorem for \(n\)-dimensional complete Riemannian manifolds with non-negative scalar curvature and \(3\leq n\leq 7\) possessing one asymptotically Euclidean end and possibly other arbitrary ends.
Recall that a Riemannian manifold, possessing at least one asymptotically Euclidean end, has \textit{non-negative ADM mass} if the Arnowitt-Deser-Misner (ADM) mass associated with this standard end is non-negative. The \textit{positive mass theorem} for a class of manifolds holds if manifolds in this class have non-negative ADM masses. The proof of the present version of the positive mass theorem (Theorem 1.2 in the article) is based on reducing it to the ``Geroch conjecture'' with arbitrary ends which asserts that for any \(n\)-dimensional manifold \(X\) and the \(n\)-dimensional torus \(T^n\) there is no smooth complete metric with positive scalar curvature on the connected sum \(X\# T^n\); moreover, the only complete metric on \(X\# T^n\) with non-negative scalar curvature is flat. Since the ``Geroch conjecture'' has been proved recently for dimensions strictly less than eight, the result in the article follows. Some consequences for asymptotically locally Euclidean spaces are also discussed (see Corollary 1.7).
Reviewer: Gábor Etesi (Budapest)Infinite families of manifolds of positive \(k\)th-intermediate Ricci curvature with \(k\) smallhttps://zbmath.org/1522.530212023-12-07T16:00:11.105023Z"Domínguez-Vázquez, Miguel"https://zbmath.org/authors/?q=ai:dominguez-vazquez.miguel"González-Álvaro, David"https://zbmath.org/authors/?q=ai:gonzalez-alvaro.david"Mouillé, Lawrence"https://zbmath.org/authors/?q=ai:mouille.lawrenceRecall that the Ricci curvature of a Riemannian \(n\)-manifold \((M^n,g)\) is said to be positive if for every point \(p \in M\) and every unit vector \(x \in T_p M\), \[ \mathrm{Ric}_g(x,x) = \sum_i g(R(x,e_i)e_i,x) = \sum_i K(x,e_i) > 0\] for any orthonormal \((n-1)\)-frame \(e_1,\dots,e_{n-1}\) perpendicular to \(x\) in \(T_p M\). The sectional curvature is positive if for every point \(p \in M\) and every unit vector \(x \in T_pM\), there holds \(g(R(x,y)y,x) = K(x,y) > 0\) for any unit vector \(y\) perpendicular to \(x\) in \(T_pM\). It is natural to consider the intermediate notion where at each point \(p \in M\) and for every unit vector \(x \in T_pM\) we have that \(\sum_i K(x,e_i) > 0\) for any orthonormal \(k\)-frame \(e_1,\dots,e_k\) perpendicular to \(x\) in \(T_pM\). In this case we say that \((M,g)\) has positive \(k\)-th intermediate Ricci curvature (or \(\mathrm{Ric}_k >0\) for short). This article deals with the existence of infinite families of such manifolds when \(k\) is small (note that when \(k=1\), one is considering manifolds of positive sectional curvature; infinite families are known to exist in dimensions 7 and 13).
The authors establish that:
(A) For \(n \geq 1\) there are infinitely many pairwise homotopically distinct closed simply-connected \((4n+3)\)-manifolds of positive \(k\)-th intermediate Ricci curvature for some \(k < (4n+3)/2\);
(B) For \(n\geq1\) there are closed simply-connected \(4n\)-manifolds that admit positive \(k\)-th intermediate Ricci curvature for some \(k<2n\) that do not admit positive sectional curvature, and some closed simply-connected \((4n+1)\)-manifolds that admit positive \(k\)-th intermediate Ricci curvature for \(k<(4n+1)/2\) that do not admit positive sectional curvature.
The authors prove (A) by showing that each generalized Aloff-Wallach space \(W^{4n+3} \equiv \mathrm{SU}(n+2)/\mathrm{S}\left(\mathrm{U}^{p,q} \times \mathrm{U}(n) \right)\) is a closed simply-connected \((4n+3)\)-manifold with positive \((2n-1)\)-th intermediate Ricci curvature if \(n \neq 2 \) and with positive fourth-intermediate Ricci curvature if \(n=2\). A routine calculation of cohomology rings shows that infinitely many homotopy types exist among the generalized Aloff-Wallach spaces in each of these dimensions.
They prove (B) by showing that the Grassmannian of 2-planes \(\mathrm{Gr}_2(\mathbb{R}^{2n+2})\) in \(\mathbb{R}^{2n+2}\) is a non-simply connected orientable \(4n\)-manifold with positive \(2n\)-th intermediate Ricci curvature and that the projectivized tangent bundle of the real projective \((2n+1)\)-space \(P_\mathbb{R}\mathbb{R}P^{2n+1}\) is a non-orientable \((4n+1)\)-manifold with positive \(2n\)-th intermediate Ricci curvature. None of them can have positive sectional curvature by Synge's theorem.
Reviewer: Owen Dearricott (Melbourne)Conjectures on convergence and scalar curvaturehttps://zbmath.org/1522.530222023-12-07T16:00:11.105023Z"Sormani, Christina"https://zbmath.org/authors/?q=ai:sormani.christinaSummary: Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on \textit{Scalar Curvature and Convergence}. We have tried to survey all the progress toward these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with non-negative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the chapter even though she is the only author listed above. In truth we are a team of over 30 people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.
For the entire collection see [Zbl 1517.53004].Geodesic coordinates for the pressure metric at the Fuchsian locushttps://zbmath.org/1522.530232023-12-07T16:00:11.105023Z"Dai, Xian"https://zbmath.org/authors/?q=ai:dai.xianThe pressure metric for Anosov representations was introduced by \textit{M. Bridgeman} et al. [Geom. Funct. Anal. 25, No. 4, 1089--1179 (2015; Zbl 1360.37078)] using the thermodynamic formalism, and entails in particular a mapping class group invariant Riemannian metric on the Hitchin components which restricts to the Weil-Petersson metric on the Teichmüller space. It is defined on the tangent space of a Hitchin component by taking the variance of the first variations of appropriately defined reparameterized geodesic flows to a Hitchin representation using Hölder functions.
Several \(C^0\)-properties of the pressure metric have been identified by \textit{F. Labourie} and \textit{R. Wentworth} [Ann. Sci. Éc. Norm. Supér. (4) 51, No. 2, 487--547 (2018; Zbl 1404.37036)] and the author of this article sets to investigate variational \(C^1\)-properties of the pressure metric by further expanding the use of tools in the thermodynamic formalism.
The results obtained in the article are about the Hitchin component \(\mathcal{H}^3(S)\) of surface group representations into the group \(\mathrm{PSL}(3,\mathbb{R})\); this space coincides with the space of convex real projective structures on the surface \(S\). The coordinates the author is choosing in order to find and evaluate expressions for the derivatives of the pressure metric in this case come from the parameterization of \(\mathcal{H}^3(S)\) using Higgs bundles, as pioneered by \textit{N. J. Hitchin} [Topology 31, No. 3, 449--473 (1992; Zbl 0769.32008)]. The main result obtained is that for any point in the Fuchsian locus this parameterization gives \textit{geodesic coordinates} for the pressure metric at this point.
Restricting attention to the case \(\mathcal{H}^3(S)\) amounts to the fact that there are two types of tangential directions, those given by quadratic differentials and those given by cubic differentials; this is derived from the Hitchin parameterization. The author then completes a careful computation of the derivatives of the metric tensor according to the different cases appearing. In particular, there are in total six types of first derivative of metric tensors that need to be treated and it is carefully shown that they all vanish. For the proof, the author develops a general method for computing first derivatives of the pressure metric using further tools from the thermodynamic formalism. The first variations of the reparameterization functions on closed geodesics are studied using a gauge-theoretic formula in [F. Labourie and R. Wentworth, loc. cit.], thus interpreting the resulting formula by means of a system of homogeneous ordinary differential equations. The second variations of these functions are, in turn, understood via the first variation of this gauge-theoretic formula.
The article stands out for a very meticulous treatment of each tangential direction along the Fuchsian locus in \(\mathcal{H}^3(S)\) and for the explicitness in the computations leading to the result that these coordinates are indeed geodesic. The method followed could be used to study the variational \(C^1\)-properties of the pressure metric for Hitchin components of higher rank, however, the analysis is expected to be more demanding there since more cases of tangential directions along the Fuchsian locus must be carefully studied. The author's expectation is that also in these higher rank cases, the Hitchin parameterization still provides geodesic coordinates for the pressure metric.
Reviewer: Georgios Kydonakis (Patras)The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvaturehttps://zbmath.org/1522.530242023-12-07T16:00:11.105023Z"Beran, Tobias"https://zbmath.org/authors/?q=ai:beran.tobias"Ohanyan, Argam"https://zbmath.org/authors/?q=ai:ohanyan.argam"Rott, Felix"https://zbmath.org/authors/?q=ai:rott.felix"Solis, Didier A."https://zbmath.org/authors/?q=ai:solis.didier-aIn the 1960s, \textit{V. A. Toponogov} proved his well-known splitting theorems for
Riemannian manifolds [Transl., Ser. 2, Am. Math. Soc. 37, 287--290 (1964; Zbl 0138.42902); Transl., Ser. 2, Am. Math. Soc. 70, 225--239 (1968; Zbl 0187.43801)]. \textit{A. D. Milka} [Ukr. Geom. Sb. 4, 43--48 (1967; Zbl 0184.47404)] generalized some of Toponogov's results to Alexandrov
spaces of non-negative curvature.
In this paper, the authors prove splitting theorems for Lorentzian length
spaces. Let \(\left( X,D\right) \) be a metric space with a reflexive and
transitive relation \(\leq\) (called causal relation), a transitive
relation \(<<\) (called time-like relation) contained in \(\leq\), and a lower
semi-continuous mapping (called time separation) \(\tau:X\times X\rightarrow
\left[ 0,+\infty\right] \) such that \(\tau\left( x,y\right) =0\) if
\(x\nleq y\), \(\tau\left( x,y\right) >0\Leftrightarrow x<<y\), \(\tau\) satisfies
the reverse triangle inequality, and for all \(x\leq y,x\neq y,\)
\[
\tau\left( x,y\right) =\sup\left\{ L_{\tau}\left( \gamma\right) \text{:
}\gamma\text{ future directed causal from }x\text{ to }y\right\}.
\]
Then \(\left( X,d,<<,\leq,\tau\right) \) is called a Lorentzian length space.
The main theorem states that if \(\left( X,d,<<,\leq,\tau\right) \) is a
connected, regularly localisable, globally hyperbolic Lorentzian length space
with proper metric \(d\) and global non-negative time-like curvature satisfying
time-like geodesic prolongation, and containing a complete time-like line
\(\gamma:\mathbb{R}\rightarrow X\), then there is a \(\tau\)- and causality-preserving homeomorphism
\(f:\mathbb{R}\times S\rightarrow X\), where \(S\) is a proper, strictly intrinsic metric space
of non-negative Alexandrov curvature.
Reviewer: Igor G. Nikolaev (Urbana)Hyperbolic angles in Lorentzian length spaces and timelike curvature boundshttps://zbmath.org/1522.530252023-12-07T16:00:11.105023Z"Beran, Tobias"https://zbmath.org/authors/?q=ai:beran.tobias"Sämann, Clemens"https://zbmath.org/authors/?q=ai:samann.clemensInspired by \textit{ E. H. Kronheimer} and \textit{R. Penrose} [Proc. Camb. Philos. Soc. 63, 481--501 (1967; Zbl 0148.46502)], a framework for Lorentzian geometry in the synthetic geometric spirit of Alexandrov spaces and CAT(\(k\))-spaces was developed in [\textit{M. Kunzinger} and \textit{C. Sämann}, Ann. Global Anal. Geom. 54, No. 3, 399--447 (2018; Zbl 1501.53057)] under the name of Lorentzian (pre-)length spaces. A notion of hyperbolic angle, of an angle between time-like curves, of a time-like tangent cone, as well as of an exponential map were introduced to provide a language in which to ask questions similar to those asked in synthetic differential geometry. The exploration leads to results such as: a triangle inequality for (upper) angles, the characterization of time-like curvature bounds (defined via triangle comparison) with an angle monotonicity condition, an improvement of a geodesic non-branching result for spaces with time-like curvature bounded below.
Reviewer: Victor V. Pambuccian (Glendale)Weakly non-collapsed RCD spaces are strongly non-collapsedhttps://zbmath.org/1522.530262023-12-07T16:00:11.105023Z"Brena, Camillo"https://zbmath.org/authors/?q=ai:brena.camillo"Gigli, Nicola"https://zbmath.org/authors/?q=ai:gigli.nicola"Honda, Shouhei"https://zbmath.org/authors/?q=ai:honda.shouhei"Zhu, Xingyu"https://zbmath.org/authors/?q=ai:zhu.xingyuAuthors' abstract: We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by \textit{G. De Philippis} and the second author [J. Éc. Polytech., Math. 5, 613--650 (2018; Zbl 1409.53038)] in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties
\begin{itemize}
\item \(\operatorname{tr}(\operatorname{Hess}f)=\Delta f\) on \(U\subseteq{\mathsf{X}}\) for every \(\mathrm{f}\) sufficiently regular,
\item \(\mathfrak{m}=c\mathscr{H}^n\) on \(U\subseteq{\mathsf{X}}\) for some \(c>0\),
\end{itemize}
where \(U\subseteq{\mathsf{X}}\) is open and \(\mathrm{X}\) is a -- possibly collapsed -- RCD space of essential dimension \(n\).
Reviewer: Mohammed El Aïdi (Bogotá)Rigidity for the logarithmic Sobolev inequality on complete metric measure spaceshttps://zbmath.org/1522.530272023-12-07T16:00:11.105023Z"Conrado, Franciele"https://zbmath.org/authors/?q=ai:conrado.francieleSummary: In this work, we study the rigidity problem for the logarithmic Sobolev inequality on a complete metric measure space \((M^n,g,f)\) with Bakry-Émery Ricci curvature satisfying \({\mathrm{Ric}}_f\ge \frac{a}{2}g\) for some \(a>0\). We prove that if equality holds, then \(M\) is isometric to \(\Sigma \times{\mathbb{R}}\) for some complete \((n-1)\)-dimensional Riemannian manifold \(\Sigma\) and by passing an isometry, \((M^n,g,f)\) must split off the Gaussian shrinking soliton \(({\mathbb{R}}, dt^2, \frac{a}{2}|\cdot |^2)\). This was proved in [Manuscr. Math. 162, No. 1--2, 271--282 (2020; Zbl 1439.53039)] by \textit{S.-i. Ohta} and \textit{A. Takatsu}. In this paper, we prove this rigidity result using a different method.The Plateau-Douglas problem for singular configurations and in general metric spaceshttps://zbmath.org/1522.530282023-12-07T16:00:11.105023Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paul"Fitzi, Martin"https://zbmath.org/authors/?q=ai:fitzi.martinThe Plateau problem asks if there is a surface of least area which spans a prescribed contour \(\Gamma\). Celebrated work by \textit{J. Douglas} [Trans. Am. Math. Soc. 33, 263--321 (1931; Zbl 0001.14102); Ann. Math. (2) 40, 205--298 (1939; Zbl 0020.37402)]
and \textit{T. Radó} [Ann. Math. (2) 31, 457--469 (1930; JFM 56.0437.02)]
gave a positive answer for surfaces in Euclidean space and contours \(\Gamma\) consisting of several disjoint rectifiable Jordan curves. Later these results were generalized in different directions by \textit{J. Jost} [Riemannian geometry and geometric analysis. 7th edition. Cham: Springer (2017; Zbl 1380.53001)] and others.
Let \(X\) be a smooth complete Riemannian manifold and \(M\) a compact orientable surface with \(k\geq1\) boundary components \(\partial M_{s}\). Let \(\Gamma\) be a collection of \(k\) closed rectifiable curves \(\Gamma_{j}\) in \(X\). A mapping \(u\) in the Sobolev class \(W^{1,2}\left( M,X\right) \) is said to span \(\Gamma\) if for each \(\Gamma_{j}\) there is a boundary component \(\partial M_{s}\) such that the trace \(u|_{\partial M_{s}}\) is a parametrization of \(\Gamma_{j}\). The authors use the notation \(\Lambda\left( M,\Gamma,X\right) \) to denote the set of \(u\in W^{1,2}\left( M,X\right) \) which span \(\Gamma\). The least area is defined by \(a\left( M,\Gamma,X\right) =\inf\left\{ \operatorname{Area}\left( u\right) \text{ : }u\in\Lambda\left( M,\Gamma,X\right) \right\} \) and \(a_{p}\left( \Gamma,X\right) =a\left( M,\Gamma,X\right) \) if \(M\) is the unique (up to a diffeomorphism) connected surface of genus \(p\) with \(k\) boundaries components.The Douglas condition is said to hold for \(p,\Gamma\) and \(X\) if \(a_{p}\left( \Gamma,X\right) \) is finite and \(a_{p}\left( \Gamma,X\right) <a\left( M,\Gamma,X\right) \) for every \(M\) as above and of one of the following types: either \(M\) is connected and of genus strictly smaller than \(p\), or \(M\) is disconnected and of total genus at most \(p\). The authors' main result states that if \(X\) is a smooth complete Riemannian manifold, \(\Gamma\) is a configuration of \(k\geq1\) closed rectifiable curves in \(X\) and \(M\) is a compact connected and orientable surface of genus \(p\geq0\) with \(k\) boundary components, such that the Douglas condition holds for \(p,\Gamma\) and \(X\), then there is \(u\in\Lambda\left( M,\Gamma,X\right) \) and a Riemannian metric \(g\) on \(M\) such that \(\operatorname{Area}\left( u\right) =a_{p}\left( \Gamma,X\right) \) and \(u\) is weakly conformal w.r.t. \(g\) on \(M\backslash u^{-1}\left( \Gamma\right) \). In particular, the authors' main result solves the Plateau-Douglas problem for potentially singular configurations \(\Gamma\) possibly consisting of self-intersectig curves.
Recall that a metric space is called proper if all closed and bounded sets in it are compact. The authors present a similar solution of the Plateau-Douglas problem when \(X\) is a proper metric space.
A solution of Plateau's problem in proper Aleksandrov spaces of curvature bounded above for a rectifiable Jordan's contour \(\Gamma\) was given by \textit{I. G. Nikolaev} [Sib. Math. J. 20, 246--252 (1979; Zbl 0434.53045)].
Reviewer: Igor G. Nikolaev (Urbana)\(d_p\)-convergence and \(\epsilon\)-regularity theorems for entropy and scalar curvature lower boundshttps://zbmath.org/1522.530292023-12-07T16:00:11.105023Z"Lee, Man-Chun"https://zbmath.org/authors/?q=ai:lee.man-chun"Naber, Aaron"https://zbmath.org/authors/?q=ai:naber.aaron"Neumayer, Robin"https://zbmath.org/authors/?q=ai:neumayer.robinAuthors' abstract: Consider a sequence of Riemannian manifolds \((M^n_i, g_i)\) whose scalar curvatures and entropies are bounded from below by small constants \(R_i, \mu_i \geq -\epsilon_i\). The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. As a first issue, even in the seemingly rigid case \(\epsilon_i\to 0\), we will construct examples showing that from the Gromov-Hausdorff or intrinsic flat points of view, such a sequence may converge wildly, in particular to metric spaces with varying dimensions and topologies and at best a Finsler-type structure. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on.
Instead, we will introduce a weaker notion of convergence called \(d_p\)-convergence, which is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces will have a well-behaved topology, measure theory and analysis. This includes the existence of gradients of functions and absolutely continuous curves, though potentially there will be no reasonably associated distance function. Under this \(d_p\) notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact always be close to Euclidean space, and this will constitute our \(\epsilon\)-regularity theorem. In particular, any sequence \((M^n_i,g_i)\) with lower scalar curvature and entropies tending to zero must \(d_p\)-converge to Euclidean space.
More generally, we have a compactness theorem saying that sequences of Riemannian manifolds \((M^n_i,g_i)\) with small lower scalar curvature and entropy bounds \({R_i,\mu_i \geq-\epsilon}\) must \(d_p\)-converge to such a rectifiable Riemannian space \(X\). In the context of the examples from the first paragraph, it may be that the distance functions of \(M_i\) are degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an \(L^\infty \)-Sobolev embedding and a~priori \(L^p\) scalar curvature bounds for \(p<1\).
Reviewer: Mehmet Küçükaslan (Mersin)Polynomial growth and asymptotic dimensionhttps://zbmath.org/1522.530302023-12-07T16:00:11.105023Z"Papasoglu, Panos"https://zbmath.org/authors/?q=ai:papasoglu.panosThe author of this paper proves that a graph of polynomial growth strictly less than \(n^{k+1}\) has asymptotic dimension at most \(k\). This result refines the fact shown by \textit{M. Bonamy} et al. [``Asymptotic dimension of minor-closed families and Assouad-Nagata dimension of surfaces'', Preprint, \url{arXiv:2012.02435}] that graphs with polynomial growth have finite asymptotic dimension. As a corollary, it is shown that Riemannian manifolds of bounded geometry with polynomial growth strictly less than \(n^{k+1}\) have asymptotic dimension at most \(k\). Furthermore, the author shows that there are graphs of growth less than \(n^{1+\epsilon}\) for any \(\epsilon >0\) which have infinite asymptotic Assouad-Nagata dimension. Finally, insights and questions are given regarding the relationship between growth functions and coarse embeddings.
Reviewer: Yutaka Iwamoto (Niihama)Souplet-Zhang and Hamilton-type gradient estimates for non-linear elliptic equations on smooth metric measure spaceshttps://zbmath.org/1522.530312023-12-07T16:00:11.105023Z"Taheri, Ali"https://zbmath.org/authors/?q=ai:taheri.ali-karimi|taheri.ali"Vahidifar, Vahideh"https://zbmath.org/authors/?q=ai:vahidifar.vahidehSummary: In this article, we present new gradient estimates for positive solutions to a class of non-linear elliptic equations involving the \(f\)-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet-Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry-Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.
{{\copyright} 2023 The Authors. \textit{Mathematika} is copyright {\copyright} University College London and published by the London Mathematical Society on behalf of University College London.}Deformed Dirac operators and scalar curvaturehttps://zbmath.org/1522.530322023-12-07T16:00:11.105023Z"Zhang, Weiping"https://zbmath.org/authors/?q=ai:zhang.weiping.1|zhang.weipingThis is a very informative survey on recent advances in the treatment of certain questions by Gromov regarding scalar curvature, using deformed Dirac operators and noncommutative geometry methods. The author first discusses the application of a relative index theorem to Gromov's questions about the infimum of the curvature of the tangent bundle. Then, scalar curvature along a (regular) foliation is discussed. An account is given on recent results by the author in this direction, using an appropriate deformed Dirac operator together with the Connes fibration associated with a foliation. The author also discusses the difficulty in proving these results with the classical differential geometry methods.
For the entire collection see [Zbl 1517.53004].
Reviewer: Iakovos Androulidakis (Athína)Local rigidity of manifolds with hyperbolic cusps. I: Linear theory and microlocal toolshttps://zbmath.org/1522.530332023-12-07T16:00:11.105023Z"Guedes Bonthonneau, Yannick"https://zbmath.org/authors/?q=ai:bonthonneau.yannick-guedes"Lefeuvre, Thibault"https://zbmath.org/authors/?q=ai:lefeuvre.thibaultThe authors consider the question of infinitesimal spectral rigidity of negatively curved manifolds, see, e.g., [\textit{V. Guillemin} and \textit{D. Kazhdan}, Topology 19, 301--312 (1980; Zbl 0465.58027)]. The paper is the first in a series of two articles whose aim is to extend to the case of negatively curved manifolds with hyperbolic cusps a recent result in [\textit{C. Guillarmou} and \textit{T. Lefeuvre}, Ann. Math. (2) 190, No. 1, 321--344 (2019; Zbl 1506.53054)] dealing with the local rigidity of the marked length spectrum in the case of compact negatively curved Riemannian manifolds. It is known that the \(L_2\)-spectrum of the Laplacian of negatively curved manifolds with hyperbolic cusps splits into a pure point spectrum -- eigenvalues -- and a continuous spectrum. Associated to the later one can define a resonance set, which leads to a further natural study of isospectrality, see [\textit{W. Müller}, Invent. Math. 109, No. 2, 265--305 (1992; Zbl 0772.58063); Math. Nachr. 111, 197--288 (1983; Zbl 0529.58035)]. For closed manifolds, the infinitesimal spectral rigidity is assured through the injectivity of the X-ray transform on divergence-free 2-tensors, meaning the integration of symmetric 2-tensors along closed geodesics. This procedure is also called solenoidal injectivity. Thus, the main results of the paper, namely, Theorem 1.1 and Corollary 1.2, concern the solenoidal injectivity of the X-ray transform in case the basis manifold is negatively curved with hyperbolic cusps. This is done by extending microlocal calculus techniques presented in [\textit{Y. G. Bonthonneau} and \textit{T. Weich}, J. Eur. Math. Soc. (JEMS) 24, No. 3, 851--923 (2022; Zbl 1495.37022)], and also further techniques introduced by the first author in [Commun. Math. Phys. 343, No. 1, 311--359 (2016; Zbl 1354.58031)]. The authors thoroughly present their methods to finally achieve, in Sections 5.5 and 5.6, the proofs of Theorem 1.1 and Corollary 1.2. The nonlinear case, namely the local marked length spectrum rigidity, is to be treated in the second part.
Reviewer: Antonio Roberto da Silva (Rio de Janeiro)Rigidity and vanishing theorems for complete translating solitonshttps://zbmath.org/1522.530342023-12-07T16:00:11.105023Z"Ha Tuan Dung"https://zbmath.org/authors/?q=ai:ha-tuan-dung."Nguyen Thac Dung"https://zbmath.org/authors/?q=ai:nguyen-thac-dung."Tran Quang Huy"https://zbmath.org/authors/?q=ai:tran-quang-huy.Summary: In this paper, we prove some rigidity theorems for complete translating solitons. Assume that the \(L^q\)-norm of the trace-free second fundamental form is finite, for some \(q\in{\mathbb{R}}\) and using a Sobolev inequality, we show that a translator must be a hyperspace. Our results can be considered as a generalization of \textit{L. Ma} and \textit{V. Miquel} [Manuscr. Math. 162, No. 1--2, 115--132 (2020; Zbl 1475.53015)], \textit{H. Wang} et al. [Pure Appl. Math. Q. 12, No. 4, 603--619 (2016; Zbl 1400.53058)], \textit{Y. L. Xin} [Calc. Var. Partial Differ. Equ. 54, No. 2, 1995--2016 (2015; Zbl 1325.53091)]. We also investigate a vanishing property for translators which states that there are no nontrivial \(L_f^p (p\ge 2)\) weighted harmonic 1-forms on \(M\) if the \(L^n\)-norm of the second fundamental form is bounded.Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacityhttps://zbmath.org/1522.530352023-12-07T16:00:11.105023Z"Mondino, Andrea"https://zbmath.org/authors/?q=ai:mondino.andrea"Templeton-Browne, Aidan"https://zbmath.org/authors/?q=ai:templeton-browne.aidanAuthors' abstract: We prove rigidity results involving the Hawking mass for surfaces immersed in a 3-dimensional, complete Riemannian manifold \((M,g)\) with non-negative scalar curvature (respectively, with scalar curvature bounded below by \(-6\)). Roughly, the main result states that if an open subset \(\Omega \subset M\) satisfies that every point has a neighbourhood \(U\subset \Omega\) such that the supremum of the Hawking mass of surfaces contained in \(U\) is non-positive, then \(\Omega\) is locally isometric to Euclidean \(\mathbb{R}^3\) (respectively, locally isometric to the Hyperbolic 3-space \(\mathbb{H}^3\)). Under mild asymptotic conditions on the manifold \((M,g)\) (which encompass as special cases the standard `asymptotically flat' or, respectively, `asymptotically hyperbolic' assumptions) the previous quasi-local rigidity statement implies a \textit{global rigidity}: if every point in \(M\) has a neighbourhood \(U\) such that the supremum of the Hawking mass of surfaces contained in \(U\) is non-positive, then \((M,g)\) is globally isometric to Euclidean \(\mathbb{R}^3\) (respectively, globally isometric to the Hyperbolic 3-space \(\mathbb{H}^3)\). Also, if the space is not flat (respectively, not of constant sectional curvature \(-1\)), the methods give a small yet explicit and strictly positive lower bound on the Hawking mass of suitable spherical surfaces. We infer a small yet explicit and strictly positive lower bound on the Bartnik mass of open sets (non-locally isometric to Euclidean \(\mathbb{R}^3\)) in terms of curvature tensors. Inspired by these results, in the appendix we propose a notion of `sup-Hawking mass' which satisfies some natural properties of a quasi-local mass.
Reviewer: Ivan C. Sterling (St. Mary's City)Anti-quasi-Sasakian manifoldshttps://zbmath.org/1522.530362023-12-07T16:00:11.105023Z"Di Pinto, D."https://zbmath.org/authors/?q=ai:di-pinto.dario"Dileo, G."https://zbmath.org/authors/?q=ai:dileo.giuliaThe authors introduce and investigate a new class of almost contact metric manifolds and call them anti-quasi-Saskian manifolds (aqS manifolds for short). Concretely, an almost contact metric manifold \((M,\varphi, \xi, \eta, g)\) with fundamental \(2\)-form \(\Phi(X,Y) = g(X,\varphi(Y))\) is said to be aqS if
\[
d\Phi = 0,\qquad N_\varphi = 2d\eta\otimes \xi,
\]
where \(N_\varphi=[\varphi,\varphi]+d\eta\otimes \xi\) and \([\varphi,\varphi]\) is the Nijenhuis torsion of \(\varphi\).
This class intersects the class of quasi-Saskian manifolds (where instead \(N_\varphi=0\) and \(d\Phi=0\)) precisely in the case of co-Kähler manifolds (i.e., where additionally \(\eta\) is closed). From the point of view of transverse geometry (with respect to the foliation by Reeb orbits) aqS geometry is characterized by a Kähler structure, together with a closed \((2,0)\)-form. The authors derive a Boothby-Wang fibration with this type of base manifold, in case the Reeb vector field is regular. As an important special case, there appear hyper-Kähler manifolds as quotients. Further examples of aqS manifolds include weighted Heisenberg groups and their nilmanifolds.
The authors show that aqS manifolds with constant sectional curvature are flat and co-Kähler, and investigate more closely aqS manifolds with \(\xi\)-sectional curvatures one. Moreover, they show the existence of a canonical metric connection with torsion on any aqS manifold and explore the consequences of their geometry.
Reviewer: Oliver Goertsches (Marburg)Motion of charged particles in a compact homogeneous Sasakian manifoldhttps://zbmath.org/1522.530372023-12-07T16:00:11.105023Z"Ikawa, Osamu"https://zbmath.org/authors/?q=ai:ikawa.osamuSummary: We shall construct a homogeneous Sasakian manifold \(\widetilde{M}\) from a Kähler \(C\)-space with its second Betti number one, and study a curvature property of \(\widetilde{M}\). In addition when \(M\) is a Kähler \(C\)-space with two isotropy summands, we concretely solve the motion of a charged particle in \(\widetilde{M}\) and show that if the motion of a charged particle intersects itself, then it is simply closed.
For the entire collection see [Zbl 1508.53008].On Einstein-type almost Kenmotsu manifoldshttps://zbmath.org/1522.530382023-12-07T16:00:11.105023Z"Kumara, Huchchappa Aruna"https://zbmath.org/authors/?q=ai:kumara.huchchappa-aruna"Praveena, Mundalamane Manjappa"https://zbmath.org/authors/?q=ai:praveena.mundalamane-manjappa"Naik, Devaraja Mallesha"https://zbmath.org/authors/?q=ai:naik.devaraja-malleshaA Riemannian manifold \((M, g)\) is called {of Einstein-type} if there exist two smooth functions \(f, \sigma : M\rightarrow \mathbb{R}\) such that the following generalization of the Einstein condition holds: \(f\mathrm{Ric}=\mathrm{Hess}_f+\sigma g\). The authors study this equation for a certain class of almost Kenmotsu manifolds \(M^{2n+1}\). In the main result they prove that \((M, g)\) is locally isometric with some warped products according to the value of \(n\). No information about the function \(\sigma \) is provided.
Reviewer: Mircea Crâşmăreanu (Iaşi)A result of rigidity for Ricci-flat warped productshttps://zbmath.org/1522.530392023-12-07T16:00:11.105023Z"Menezes, Ilton"https://zbmath.org/authors/?q=ai:menezes.ilton"Correia, Paula"https://zbmath.org/authors/?q=ai:correia.paula"Pina, Romildo"https://zbmath.org/authors/?q=ai:pina.romildo-da-silva\textit{A. Einstein} [Ann. Math. (2) 40, 922--936 (1939; Zbl 0023.42501)] proposed a field equation to describe the gravitational effects produced by a given mass in general relativity,
\[
\mathrm{Ric}_g -\frac{1}{2}sg=T,
\]
where \(g\) is a Lorentz metric on a manifold \(M^4\), \(s\) is the scalar curvature of \((M^4, g)\) and \(T\) is the stress/energy tensor, a symmetric 2-covariant tensor field on \(M\). This equation is known as the Einstein field equation, see, e.g., [\textit{A. L. Besse}, Einstein manifolds. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)]. When \(T \equiv 0\), we obtain an equation for a gravitational field in a vacuum and in this case the metric \(g\) satisfies the condition
\[
\mathrm{Ric}_g =\lambda g,\, \, \lambda \in \mathbb R.
\]
Semi-Riemannian manifolds \((M^n, g)\) that satisfy the last equation are said to be Einstein. In particular, when \(\lambda = 0, (M^n, g)\) is said to be Ricci-flat. The classical solution for the Einstein field equation in a vacuum is the Schwarzschild metric: it models the gravitational fields outside an isolated, static, spherically symmetric star. An Einstein metric is a good candidate for a privileged metric on a given manifold. In dimension two, the only Einstein manifolds are space forms, which have constant curvature. In dimensions greater than two, having constant Ricci curvature is a good condition, representing an intermediate level between constant sectional curvature and constant scalar curvature. Einstein manifolds are special in different contexts: they are fixed points of the famous Ricci flow and Yamabe flow, and critical points of Riemannian functionals, among others. There are many studies on such manifolds and the interest in obtaining new classifications and examples extends to the present day. Conforming transformations between Einstein spaces were studied in [\textit{W. Kühnel}, Aspects Math.: E, 12, 105--146 (1988; Zbl 0667.53039)], where, in particular, it was shown that locally conformally flat Einstein manifolds have constant sectional curvature. This result was generalized to the semi-Riemannian case in [\textit{W. Kühnel} and \textit{H. B. Rademacher}, Proc. Am. Math. Soc. 123, No. 9, 2841--2848 (1995; Zbl 0851.53039)]. So, we have no non-trivial Einstein manifolds with metric conformal to a pseudo-Euclidean metric. A good strategy to obtain locally conformally flat manifolds is to study warped products of two semi-Riemannian manifolds \((B^n, g_B)\) and \((F^m, g_F )\). The manifold \(B\) is said to be base, \(F\) is the fiber and \(h\) is the warping function, and the warped product is denoted by \(B\times_h F\).
Generalized Schwarzschild metrics invariant by rotation were obtained by
\textit{J. P. dos Santos} and \textit{B. Leandro} [J. Math. Anal. Appl. 469, No. 2, 882--896 (2019; Zbl 1400.83020)].
They studied the static vacuum Einstein equation and by using an Ansatz method, they found the most general invariant that reduces the system of PDEs which describes the problem to a system of ODEs. The invariant found by them shows that the maximum invariance is obtained by rotation. They also provided the entire set of solutions of the reduced system. They studied Einstein warped products when the base is conformal to a pseudo-Euclidean space and presented the system of PDEs that describes the problem, thus obtaining all Ricci-flat solutions invariant under the action of the \((n- 1)\)-dimensional translation group.
In this work the authors seek solutions for a system of PDEs invariant by rotation, i.e., when the functions assume the same value across \((n-1)\)-dimensional spheres in \(\mathbb R^n\), transforming the system of PDEs into a system of ordinary differential equations.
They prove a rigidity result on Ricci-flat semi-Riemannian warped products when the base is locally conformally flat with scalar curvature zero and invariant by the action of the pseudo-orthogonal group. They find that in these manifolds the fiber must be \(1\)-dimensional. As an application, every metric with these characteristics is a generalized Schwarzschild metric.
Reviewer: Cenap Özel (Jeddah)Kenmotsu 3-manifolds and gradient solitonshttps://zbmath.org/1522.530402023-12-07T16:00:11.105023Z"Mofarreh, F."https://zbmath.org/authors/?q=ai:mofarreh.fatemah-y-y"De, U. C."https://zbmath.org/authors/?q=ai:de.uday-chandThe paper is about three-dimensional Kenmotsu metrics which are gradient Yamabe soliton or gradient Einstein soliton. An example of Kenmotsu metric having the same potential function and with both gradient Yamabe and gradient Einstein solitons is discussed.
Reviewer: Mircea Crâşmăreanu (Iaşi)Magnetic curves in quasi-Sasakian manifolds of product typehttps://zbmath.org/1522.530412023-12-07T16:00:11.105023Z"Munteanu, Marian Ioan"https://zbmath.org/authors/?q=ai:munteanu.marian-ioan"Nistor, Ana Irina"https://zbmath.org/authors/?q=ai:nistor.ana-irinaIn a previous work [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 214, Article ID 112571, 18 p. (2022; Zbl 1485.53033)] the authors stated the following conjecture: A contact magnetic curve in a quasi-Sasakian manifold of dimension \(2n+1\geq 5\) must be a Frenet helix of maximum order \(5\). The paper gives a positive answer to this conjecture. The main focus is on magnetic curves in a quasi-Sasakian manifold of product type \(N^{2p+1}\times B^{2k}\), where \(N\) is a Sasakian manifold and \(B\) is a Kähler one. Concrete examples of magnetic curves are provided for the manifold \(\mathbb{S}^3\times \mathbb{S}^2\).
For the entire collection see [Zbl 1508.53008].
Reviewer: Mircea Crâşmăreanu (Iaşi)Some geometric properties of \(\eta_\ast \)-Ricci solitons on \(\alpha \)-Lorentzian Sasakian manifoldshttps://zbmath.org/1522.530422023-12-07T16:00:11.105023Z"Pandey, Shashikant"https://zbmath.org/authors/?q=ai:pandey.shashikant"Singh, Abhishek"https://zbmath.org/authors/?q=ai:singh.abhishek.2"Prasad, Rajendra"https://zbmath.org/authors/?q=ai:prasad.rajendraAuthors' abstract:
``We investigate the geometric properties of \(\eta_\ast \)-Ricci solitons on \(\alpha \)-Lorentzian Sasakian \(( \alpha \)-LS) manifolds, and show that a Ricci semisymmetric \(\eta_\ast \)-Ricci soliton on an \(\alpha \)-LS manifold is an \(\eta_\ast \)-Einstein manifold. Further, we study \(\varphi_\ast \)-symmetric \(\eta_\ast \)-Ricci solitons on such manifolds. We prove that \(\varphi_\ast \)-Ricci symmetric \(\eta_\ast \)-Ricci solitons on an \(\alpha \)-LS manifold are also \(\eta_\ast \)-Einstein manifolds and provide an example of a 3-dimensional \(\alpha \)-LS manifold for the existence of such solitons.''
Reviewer's remark:
There is no explanation for the use of the symbol \(\ast\) throughout the paper. Unfortunately, although there are 30 references, there is only one example in dimension~\(3\).
Reviewer: Mircea Crâşmăreanu (Iaşi)A note on Legendre trajectories on Sasakian space formshttps://zbmath.org/1522.530432023-12-07T16:00:11.105023Z"Shi, Qingsong"https://zbmath.org/authors/?q=ai:shi.qingsong"Adachi, Toshiaki"https://zbmath.org/authors/?q=ai:adachi.toshiakiSummary: In this paper we give a report on the distribution of lengths of Legendre trajectories on complete simply connected Sasakian space forms following to [\textit{T. Adachi}, J. Geom. 90, No. 1--2, 1--29 (2008; Zbl 1176.53072)].
For the entire collection see [Zbl 1508.53008].Twistor theory for exceptional holonomyhttps://zbmath.org/1522.530442023-12-07T16:00:11.105023Z"Pantilie, Radu"https://zbmath.org/authors/?q=ai:pantilie.raduSummary: We show that the \(G_2\)-manifolds and certain Spin(7)-manifolds are endowed with natural Riemannian twistorial structures. Along the way, the exceptional holonomy representations are reviewed and other related facts are considered.
{{\copyright} 2020 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence.}The \(\hat{G}\)-index of a spin, closed, hyperbolic manifold of dimension 2 or 4https://zbmath.org/1522.530452023-12-07T16:00:11.105023Z"Ratcliffe, John G."https://zbmath.org/authors/?q=ai:ratcliffe.john-g"Tschantz, Steven T."https://zbmath.org/authors/?q=ai:tschantz.steven-tLet \(M = \Gamma \backslash H^n\) be a spin, closed, hyperbolic \(n\)-manifold. This means that the discrete subgroup \(\Gamma\) of \(\mathrm{SO}^+(n,1)\) lifts to a subgroup \(\hat\Gamma\) of \(\mathrm{Spin}^+(n,1)\). Denote by \(\hat{G}\) the (finite) group of symmetries of the spin structure \(\hat\Gamma\backslash \mathrm{Spin}^+(n,1)\) of \(M\). If \(n\) is even, then \(\hat G\) acts on the finite-dimensional, complex vector spaces \(\mathcal{H}^+\) and \(\mathcal{H}^-\) of positive and negative harmonic spinors on \(M\). This gives two representations \(\rho^+\) and \(\rho^-\) of \(\hat G\) whose difference \(\rho^+ - \rho^-\) in the representation ring \(R(\hat G)\) is the \(\hat G\)-index of \(M\).
In the present paper, general techniques are developed to compute this index for \(n=2\) and \(n=4\), and they are applied to compute it for the (unique) fully symmetric spin structure of the Davis hyperbolic 4-manifold. The main workhorse is a representation of \(\mathrm{Spin}^+(4,1)\) by \(\mathrm{SU}(1,1;\mathbb{H})\) which allows the authors to compute the difference of the characters of the two representations \(\rho^+\) and \(\rho^-\) of \(\hat G\) (Theorem 7.5 in the paper).
Reviewer: Alexander Engel (Greifswald)The topology of compact rank-one ECS manifoldshttps://zbmath.org/1522.530462023-12-07T16:00:11.105023Z"Derdzinski, Andrzej"https://zbmath.org/authors/?q=ai:derdzinski.andrzej"Terek, Ivo"https://zbmath.org/authors/?q=ai:terek.ivoSummary: Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution \(\mathcal{D} \). All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of \(\mathcal{D}^\perp\) serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that \(\,\mathcal{D}^\perp\) has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of \(\mathcal{D}^\perp\) are the factor manifolds of a global product decomposition.Geometric properties of non-flat totally geodesic surfaces in symmetric spaces of type Ahttps://zbmath.org/1522.530472023-12-07T16:00:11.105023Z"Ohashi, Misa"https://zbmath.org/authors/?q=ai:ohashi.misa"Suzuki, Kazuhiro"https://zbmath.org/authors/?q=ai:suzuki.kazuhiroSummary: The purpose of this paper is to prove that each non-flat totally geodesic surface \(S^2\) in symmetric spaces \(\mathrm{SU} (4) / \mathrm{SO}(4)\), \(\mathrm{SU}(8) / \mathrm{Sp}(4)\) or \(\mathrm{SU} (4) / S(\mathrm{U} (2) \times \mathrm{U} (2))\) can be considered as a non-flat totally geodesic surface in \(\mathrm{Spin}(5) / \mathrm{U} (2)\) which is a totally geodesic submanifold in one of the above three symmetric spaces. This non-flat totally geodesic surface is also a totally real surface with respect to the complex structure on \(\mathrm{Spin}(5) / \mathrm{U} (2) \cong Q^3\).
For the entire collection see [Zbl 1508.53008].Theta series and generalized special cycles on Hermitian locally symmetric manifoldshttps://zbmath.org/1522.530482023-12-07T16:00:11.105023Z"Shi, Yousheng"https://zbmath.org/authors/?q=ai:shi.youshengSummary: We study generalized special cycles on Hermitian locally symmetric spaces \(\Gamma \setminus D\) associated to the groups \(G=\mathrm{U}(p,q), \mathrm{Sp}(2n, \mathbb{R})\) and \(\mathrm{O}^*(2n)\). These cycles are algebraic and covered by symmetric spaces associated to subgroups of \(G\) which are of the same type. We show that Poincaré duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to \(G\) to vector-valued automorphic functions on the groups \(G'=\mathrm{U}(m,m), \mathrm{O}(m,m)\) or \(\mathrm{Sp}(m,m)\) which forms a reductive dual pair with \(G\).Minimal submanifolds in \(\mathrm{Sol}_0^4\)https://zbmath.org/1522.530492023-12-07T16:00:11.105023Z"Erjavec, Zlatko"https://zbmath.org/authors/?q=ai:erjavec.zlatko"Inoguchi, Jun-ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichiThe authors study several minimal submanifolds in the 4-dimensional homogeneous solvable Lie group Sol\(_0^4\) equipped with the standard globally conformal Kähler structure. They begin with the classification of minimal proper invariant submanifolds in Sol\(_0^4\) and show that those which are tangent or normal to the Lee field are cylindrical surfaces. The also find minimal CR-products in Sol\(_0^4\).
Reviewer: Ernest L. Stitzinger (Raleigh)On the relationships between Hopf fibrations and Cartan hypersurfaces in sphereshttps://zbmath.org/1522.530502023-12-07T16:00:11.105023Z"Hashimoto, Hideya"https://zbmath.org/authors/?q=ai:hashimoto.hideyaSummary: The famous theorem due to Hurwitz stated that the normed (division) algebra is isomorphic to one of the following four algebras; the field \(\mathbb{R}\) of real numbers, the field \(\mathbb{C}\) of complex numbers, the algebra \(\mathbb{H}\) of quaternions, and the non-associative algebra \(\mathbb{O}\) of octonions. By using these algebraic structures, we can construct the Hopf fibrations, and Cartan hypersurfaces in a sphere. The purpose of this paper is to give the relationship between these Hopf fibrations and Cartan hypersurfaces.
For the entire collection see [Zbl 1508.53008].Biharmonic PNMCV submanifolds in Euclidean 5-spacehttps://zbmath.org/1522.530512023-12-07T16:00:11.105023Z"Şen, Rüya"https://zbmath.org/authors/?q=ai:sen.ruya"Turgay, Nurettin Cenk"https://zbmath.org/authors/?q=ai:turgay.nurettin-cenkA submanifold (or an isometric immersion) is said to have parallel normalized mean curvature vector field if the mean curvature vector field is nowhere zero and the unit mean curvature vector field is parallel in the normal bundle of the submanifold. The paper investigates 3-dimensional biharmonic submanifolds with parallel normalized mean curvature vector (PNMCV) in 5-dimensional Euclidean space. Among other things, the authors prove that there is no proper 3-dimensional biharmonic submanifold with PNMCV in 5-dimensional Euclidean space, which verifies Chen's conjecture for biharmonic submanifolds for another special case. Recall that Chen's conjecture states that any biharmonic submanifold in a Euclidean space is minimal which is equivalent to saying that there is no proper biharmonic submanifold in a Euclidean space.
Reviewer: Ye-Lin Ou (Commerce)Rigidity of Willmore submanifolds and extremal submanifolds in the unit spherehttps://zbmath.org/1522.530522023-12-07T16:00:11.105023Z"Yang, Deng-Yun"https://zbmath.org/authors/?q=ai:yang.dengyun"Fu, Hai-Ping"https://zbmath.org/authors/?q=ai:fu.haiping"Zhang, Jin-Guo"https://zbmath.org/authors/?q=ai:zhang.jinguoSummary: Let \(M\) be an \(n\)-dimensional \((n\ge 4)\) compact Willmore (or extremal) submanifold in the unit sphere \(S^{n+p}\). Denote by \({\mathrm{Ric}}\) and \(H\) the Ricci curvature and the mean curvature of \(M\), respectively. It is proved that if \((\int_M ({\mathrm{Ric}}_-^{\lambda})^\frac{n}{2})^\frac{2}{n}<A(n,\lambda ,H,\rho)\) (or \(B(n,\lambda ,H,\rho)\)), then \(M\) is a totally umbilical sphere, where \(A(n,\lambda ,H,\rho)\) and \(B(n,\lambda ,H,\rho)\) are two explicit positive constants depending on \(n\), \(\lambda\), \(H\), and \(\rho\). This extends parts of the results from a pointwise Ricci curvature lower bound to an integral Ricci curvature lower bound.Biharmonic hypersurfaces with recurrent operators in the Euclidean spacehttps://zbmath.org/1522.530532023-12-07T16:00:11.105023Z"Abedi, Esmaiel"https://zbmath.org/authors/?q=ai:abedi.esmaiel"Mosadegh, Najma"https://zbmath.org/authors/?q=ai:mosadegh.najmaThe paper studies Chen's conjecture on biharmonic submanifolds for the case of hypersurfaces. The main results obtained in the paper prove that a biharmonic hypersurface with recurrent Ricci, Riemannian, or Weyl curvature operator in a Euclidean space is minimal. Here, a tensor (or an operator) \(T\) is recurrent if there exists a 1-form \(\omega\) such that \(\nabla _XT=\omega (X)T\) for any vector field \(X\). It turns out that biharmonicity and the recurrent curvature operator assumption imply that the hypersurface has at most two distinct principal curvatures.
Reviewer: Ye-Lin Ou (Commerce)Curvature locus of a 3-manifold having a normal bundle with some flat directionhttps://zbmath.org/1522.530542023-12-07T16:00:11.105023Z"Monterde, J."https://zbmath.org/authors/?q=ai:monterde.juanThe author discusses geometric aspects of the second fundamental form of 3-manifolds in \(\mathbb R^{3+3}\) with not totally flat normal bundle. With a convenient choice of bases for the tangent space and one for the normal space, he classifies their curvature loci based on a result in [\textit{B. Riul} et al., ``Curvature loci of 3-manifolds'', Preprint, \url{arXiv:2204.12312}]. Among all the possible substantial (non-planar) curvature loci, it is shown that for this kind of immersions, Cross-Cap surfaces, Steiner type 6 surfaces or ellipsoids can not be a curvature locus. The only possibilities are Steiner Roman surfaces, Steiner type 5 surfaces or truncated cones. Furthermore, the curvature locus must admit a projection on a triangle along some direction in the normal space. This is possible for truncated cones, but not for the other two types of allowed curvature loci. Based on this, the author gives the name \textit{Steiner Roman (resp., type 5) surfaces with parallel sides} to the subfamily of Steiner Roman (resp., Steiner type 5) surfaces for which there is a projection on a triangle.
Reviewer: Maria Aparecida Soares Ruas (São Carlos)Non-positively curved Ricci surfaces with catenoidal endshttps://zbmath.org/1522.530552023-12-07T16:00:11.105023Z"Zang, Yiming"https://zbmath.org/authors/?q=ai:zang.yimingSummary: A Ricci surface is defined to be a Riemannian surface \(({\boldsymbol{M}},{\boldsymbol{g}}_{\boldsymbol{M}})\) whose Gauss curvature \({\boldsymbol{K}}\) satisfies the differential equation \({\boldsymbol{K}}\boldsymbol{\Delta} {\boldsymbol{K}} + {\boldsymbol{g}}_{\boldsymbol{M}}\left({{\mathbf{d}}{\boldsymbol{K}}},{{\mathbf{d}}{\boldsymbol{K}}}\right) + {\mathbf{4}}{\boldsymbol{K}}^{\mathbf{3}}={\mathbf{0}}\). In the case where \({\boldsymbol{K}}<{\mathbf{0}}\), this equation is equivalent to the well-known Ricci condition for the existence of minimal immersions in \({\mathbb{R}}^3\). Recently, \textit{A. Moroianu} and \textit{S. Moroianu} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 4, 1093--1118 (2015; Zbl 1334.49127)] proved that a Ricci surface with non-positive Gauss curvature admits locally an isometric minimal immersion into \({\mathbb{R}}^3\). In this paper, we are interested in studying non-compact orientable Ricci surfaces with non-positive Gauss curvature. Firstly, we give a definition of catenoidal end for non-positively curved Ricci surfaces. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data to obtain some classification results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. Furthermore, we also give an existence result for non-positively curved Ricci surfaces of arbitrary positive genus which have finite catenoidal ends.Degeneration of globally hyperbolic maximal anti-de Sitter structures along rayshttps://zbmath.org/1522.530562023-12-07T16:00:11.105023Z"Tamburelli, Andrea"https://zbmath.org/authors/?q=ai:tamburelli.andreaThis paper is about the degenerations of some geometric quantities of globally hyperbolic maximally Cauchy-compact anti-de Sitter manifolds along rays of cotangent vectors in the Krasnov-Schlenker parametrization.
Recall that anti-de Sitter space is the Lorentzian analog of hyperbolic space. A 3-dimensional manifold \(M\) is called anti-de Sitter if it is locally isometric to \(\mathrm{AdS}_3\). It is globally hyperbolic maximally Cauchy-compact (GHMC) if it contains an embedded closed, orientied surface \(S\) that intersects every inextensible causal curve in exactly one point together with a natural maximality condition. Such a manifold is diffeomorphic to \(S\times \mathbb{R}\). These manifolds are the Lorentzian analogs of almost-Fuchsian manifolds.
Two parametrizations of the set of GHMC anti-de Sitter manifolds are fundamental: the Mess parametrization by \(\mathrm{Teich}(S)\times \mathrm{Teich}(S)\) and the Krasnov-Schlenker parametrization by \(T^*\mathrm{Teich}(S)\), where \(\mathrm{Teich}(S)\) denotes the Teichmüller space of \(S\). The key point for the second parametrization is the existence of a unique embedded maximal surface \(\Sigma\) inside \(M\). It is uniquely determined by the induced metric and the second fundamental form which is described by a holomorphic quadratic differential \(q\).
The author studies three geometric quantities related to the manifold \(M\) with holonomy \(\rho\). The first is the Hölder exponent of the unique homeomorphism of \(\mathbb{RP}^1\) intertwining the two actions of \(\pi_1(S)\) on \(\mathbb{RP}^1\) coming from the Mess parametrization. The second is the Lorentzian Hausdorff dimension of the limit set of \(\rho\), recently defined by \textit{O. Glorieux} and \textit{D. Monclair} [Int. Math. Res. Not. 2021, No. 18, 13661--13729 (2021; Zbl 1487.53085)]. The last is the width of the convex core of \(M\), which is the quotient of the convex hull of the limit set by \(\rho\).
The author considers a family of manifolds \(M_t\) with Krasnov-Schlenker parametrization \((h,tq)\in T^*\mathrm{Teich}(S)\) where \(t\to\infty\) and proves that:
(1) The Hölder exponent of the limit curve tends to zero (Theorem 2.7);
(2) The Lorentzian Hausdorff dimension of the limit set tends to zero (Theorem 3.13);
(3) The width of the convex core tends to \(\pi/2\) (Proposition 4.3).
As a related result, the induced metric \(I_t\) on the maximal surface \(\Sigma_t\) is studied. From explicit upper and lower bounds it follows that \(I_t/t\) tends to the metric \(\lvert q\rvert\) outside of the zeros of \(q\) and monotonically from above (Proposition 3.10).
The paper is thoroughly written and gives many details.
Reviewer: Alexander Thomas (Heidelberg)On generalized biconservative spacelike surfaces in Lorentz space formshttps://zbmath.org/1522.530572023-12-07T16:00:11.105023Z"Yang, Dan"https://zbmath.org/authors/?q=ai:yang.dan"Zhao, ZiMin"https://zbmath.org/authors/?q=ai:zhao.ziminThe present paper is devoted to general biconservative surfaces (GB surfaces) in a Lorentzian 3-space form \(N_1^3(c)\) with constant curvature \(c\). In the case of \(c=1\) and \(c=-1\) the Lorentzian space form is the de Sitter space \(\mathbb S_1^3\) and the anti-de Sitter space \(\mathbb H_1^3\), respectively. The surfaces in consideration are characterized by the equation
\begin{center} \(A\nabla H + k H\nabla H = 0\). \end{center}
Here \(A\) and \(H\) denote the shape operator and the mean curvature, respectively; \(k\) is a constant. Obviously all CMC surfaces fulfill the above equation. The authors give a complete classification and obtain nine kinds of GB surfaces in de Sitter space \(\mathbb S_1^3\) and seven kinds of GB surfaces in anti-de Sitter space \(\mathbb H_1^3\), respectively, which are not CMC surfaces. All these solutions are surfaces of revolution. In the case of Minkowski 3-space \(N_1^3(0)=\mathbb E_1^3\) the complete explicit classification of GB surfaces was provided by \textit{D. Yang} and \textit{X. Y. Zhu} [J. Math. Anal. Appl. 496, No. 1, Article ID 124799, 18 p. (2021; Zbl 1459.53032)].
Reviewer: Friedrich Manhart (Wien)Kähler submanifolds of the real hyperbolic spacehttps://zbmath.org/1522.530582023-12-07T16:00:11.105023Z"Chion, Sergio"https://zbmath.org/authors/?q=ai:chion.sergio-j"Dajczer, Marcos"https://zbmath.org/authors/?q=ai:dajczer.marcosSummary: The local classification of Kaehler submanifolds \(M^{2n}\) of the hyperbolic space \(\mathbb{H}^{2n+p}\) with low codimension \(2\leq p\leq n-1\) under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere \(\mathbb{S}^{2n+p}\), \(2\leq p\leq n-1\), since \textit{L. A. Florit} et al. [J. Eur. Math. Soc. (JEMS) 7, No. 1, 1--11 (2005; Zbl 1090.53010)] have shown that the codimension has to be \(p=n-1\) and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in \(\mathbb{S}^{3n-1}\subset\mathbb{R}^{3n} \). The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result in [Florit et al., loc. cit.] for spherical submanifolds. Besides, we generalize several results obtained by the second author and \textit{Th. Vlachos} [Proc. Am. Math. Soc. 148, No. 9, 4015--4024 (2020; Zbl 1444.53019)].An Obata-type characterization of doubly-warped product Kähler manifoldshttps://zbmath.org/1522.530592023-12-07T16:00:11.105023Z"Ginoux, Nicolas"https://zbmath.org/authors/?q=ai:ginoux.nicolas"Habib, Georges"https://zbmath.org/authors/?q=ai:habib.georges"Pilca, Mihaela"https://zbmath.org/authors/?q=ai:pilca.mihaela"Semmelmann, Uwe"https://zbmath.org/authors/?q=ai:semmelmann.uweThe authors study complete Kähler manifolds \((\widetilde{M},\widetilde{g},J)\), with \(\dim \widetilde{M}=2n\), admitting a function \(u\) without critical points, such that the Hessian \(H^u\) is \(J\)-invariant, \(\nabla u\) is (pointwise) an eigenvector of \(H^u\) and the space \(\{\nabla u, J\nabla u\}^{\perp}\) is (pointwise) an eigenspace of \(H^u\) with eigenvalue \(\mu\). They show that \(\mu\) vanishes identically or is everywhere different from 0. In both cases the authors describe \(\widetilde{M}\). If \(\mu\ne0\), it is as a doubly warped product and \(\widetilde{M}=\mathbb R\times M\), where \(M\) is a level hypersurface of \(u\). The metric is \(dt^2+\rho(t)^2(\rho'(t)^2g_{\xi}+g_{\xi^{\perp}})\) and \((M,\xi,g)\) is Sasaki if \(n>2\) or \((M,\xi,g)\) is a minimal Riemannian flow that is basic conformally Sasaki if \(n=2\). We also have \(\rho=\sqrt{u}\). If \(\mu=0\) then \(M=\mathbb R\times\mathbb R\times\Sigma\) with the metric \(dt^2+\rho(t)^2ds^2+g_{\Sigma}\). Also such Einstein manifolds are classified.
Note that some similar results in the compact case were obtained in [the reviewer, Differ. Geom. Appl. 46, 119--131 (2016; Zbl 1337.53087)].
Reviewer: Włodzimierz Jelonek (Kraków)Special Hermitian metrics on Oeljeklaus-Toma manifoldshttps://zbmath.org/1522.530602023-12-07T16:00:11.105023Z"Otiman, Alexandra"https://zbmath.org/authors/?q=ai:otiman.alexandraSummary: Oeljeklaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension \(K\) of \(\mathbb{Q}\) and a subgroup of units \(U\). We characterize the existence of \textit{pluriclosed} metrics (also known as \textit{strongly Kähler with torsion (SKT)} metrics) on any OT manifold \(X(K, U)\) purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number \(b_3\) and the Dolbeault cohomology group \(H^{2,1}_{\overline{\partial}}\). Combined with the main result in [\textit{A. Dubickas}, Result. Math. 76, No. 2, Paper No. 78, 12 p. (2021; Zbl 1472.11275)], these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type \((2, 2)\), the existence of a pluriclosed metric on \(X(K, U)\) is entirely topological, namely, it is equivalent to \(b_3=2\). Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric. Finally, we show that no OT manifold admits \textit{balanced metrics}, but all of them carry instead \textit{locally conformally balanced metrics}.The Demailly system for a direct sum of ample line bundles on Riemann surfaceshttps://zbmath.org/1522.530612023-12-07T16:00:11.105023Z"Pingali, Vamsi Pritham"https://zbmath.org/authors/?q=ai:pingali.vamsi-prithamLet \(X\) be a projective manifold equipped with a Kähler metric \(\omega_0\), and \(E\) be a holomorphic vector bundle of rank \(r\) on \(X\) equipped with a smooth Hermitian metric \(h_0\). A conjecture of Griffiths states that \(E\) being Hartshorne ample (an algebraic/sheaf-theoretic notion) implies, and thus is equivalent to, \(E\) being Griffiths positive (a metric/differential-geometric notion). \textit{J.-P. Demailly} proposes in [Sb. Math. 212, No. 3, 305--318 (2021; Zbl 1464.32028); translation from Mat. Sb. 212, No. 3, 39--53 (2021)] an approach to solve the conjecture by solving a \(1\)-parameter family (with parameter \(t \in [0,1]\)) of systems of equations consisting of a ``matrix Monge-Ampère equation'' and an elliptic differential system of Hermitian-Yang-Mills type. See also [\textit{J.-P. Demailly}, ``Monge-Ampère functionals for the curvature tensor of a holomorphic vector bundle'', Preprint, \url{arXiv:2112.14463}] for a modified approach in the same line.
This article investigates the special case \(\dim X = 1\), when the system in [\textit{J.-P. Demailly}, Sb. Math. 212, No. 3, 305--318 (2021; Zbl 1464.32028); translation from Mat. Sb. 212, No. 3, 39--53 (2021)] is reduced to
\[
\left\{ \begin{aligned} &\det\left(\frac{\sqrt{-1} F_t}{\omega_0} +(1-t) \alpha_0\operatorname{id}_E\right) = e^{\lambda f} a_t \\
&\sqrt{-1} F_t -\frac 1r \:\operatorname{tr}\!\left(\sqrt{-1}F_t \right) \operatorname{id}_E = -e^{f_t} \ln g_t \omega_0 \end{aligned} \right. \quad\text{ for } 0\leq t \leq 1 \; ,
\]
where \(\alpha_0, \lambda \gg r\) are constants, \(a_t\) is a positive smooth function, \(\operatorname{id}_E\) is the identity map on \(E\), and \(\sqrt{-1}F_t\) is the curvature of the metric \(h_t = e^{-f_t} g_t h_0\) on \(E\), with \((f_t, g_t)\) being the unknown of the system, where \(f_t\) is a function on \(X\), and \(g_t\) is a positive-definite \(h_0\)-Hermitian endomorphism on \(E\) with \(\det g_t = 1\). It is shown that, if either
(1) \((E, h_0) = \bigoplus_{i=1}^r (L_i, h_{0\:i})\), where each \(L_i\) is an ample line bundle (but \(h_{0\:i}\) need not be positively curved), or
(2) Every \(\mathcal C^{2,\gamma}\) (\(0< \gamma < 1\)) solution \((f_t, g_t)\) of the above system satisfies \(f_t \geq -C\) for some constant \(C\) independent of \(t\),
then there exists a smooth metric \(h_t = e^{-f_t} g_t h_0\) on \(E\) whose curvature \(\sqrt{-1}F_t\) is Griffiths positive for all \(t \in [0,1]\).
The existence of a solution at \(t=0\), namely, \((f_0, g_0) = (0, g_0)\), is guaranteed by a result of \textit{K. Uhlenbeck} and \textit{S. T. Yau} [Commun. Pure Appl. Math. 39, S257--S293 (1986; Zbl 0615.58045)]. While the continuity method seems insufficient to provide solutions of the Demailly system for all \(t \in [0,1]\), the author resorts to the Leray-Schauder degree theory for the existence of solutions on the whole interval. The a priori estimate (\(f_t \geq -C\) for \((f_t,g_t)\) being \(\mathcal C^{2,\gamma}\)) is shown to hold true when \((E,h_0)\) is a direct sum of Hermitian line bundles.
Reviewer: Tsz On Mario Chan (Taipei)Mabuchi geometry of big cohomology classeshttps://zbmath.org/1522.530622023-12-07T16:00:11.105023Z"Xia, Mingchen"https://zbmath.org/authors/?q=ai:xia.mingchenLet \((X,\omega)\) be a compact Kähler manifold, and \(\mathcal{H} (X, \omega)\) be the space of smooth Kähler potentials with respect to the reference metric \(\omega\). The Finsler metric \(d_p\) on \(\mathcal{H} (X, \omega)\), introduced by \textit{T. Darvas} [Adv. Math. 285, 182--219 (2015; Zbl 1327.53093)], is defined in terms of a \(C^{1,1}\)-solution of a homogeneous complex Monge-Ampère equation, and extends by continuity to the completion \(\mathcal{E}^p (X, \omega )\) of \(\mathcal{H}(X,\omega)\). This metric, particularly when \(p=1\), plays a fundamentally important role in the study of canonical Kähler metrics.
The paper gives a new alternative definition for the \(d_p\)-metric, in terms of length segments. This new definition does not rely on the solution of the homogeneous Monge-Ampère equation, and can be extended to the case when the Kähler class \([ \omega ]\) is merely big. The \(d_p\)-metric for the big cohomology class thus constructed is shown to be complete if \([ \omega ]\) is further assumed to be nef or when \(p=1\). Moreover, this definition is birationally invariant.
The \(d_p\)-metrics for big cohomology classes were studied some earlier papers: the case \(p=1\) was done by \textit{T. Darvas} et al. [Ann. Inst. Fourier 68, No. 7, 3053--3086 (2018; Zbl 1505.53081)], the case when \([ \omega ]\) is big and nef was done by \textit{E. Di Nezza} and \textit{C. H. Lu} [Acta Math. Vietnam. 45, No. 1, 53--69 (2020; Zbl 1436.53050)], and \textit{A. Trusiani} [J. Geom. Anal. 32, No. 2, Paper No. 37, 37 p. (2022; Zbl 1487.32179)] also proved results that are closely related. The author proves that his definition generalises all these previous ones.
An envelope construction plays an important role in defining the \(d_p\)-metric in this paper. The author finds that its key properties can be abstracted axiomatically as an algebraic structure which he calls a rooftop structure, which is discussed in details in section 3 of the paper. He proves that \(\mathcal{E}^p (X, \omega )\) is a \(p\)-strict locally complete rooftop metric space, whose precise meaning is also explained in Section 3.
Reviewer: Yoshinori Hashimoto (Osaka)Some rigidity results on complete Finsler manifoldshttps://zbmath.org/1522.530632023-12-07T16:00:11.105023Z"Raeisi-Dehkordi, Hengameh"https://zbmath.org/authors/?q=ai:dehkordi.hengameh-raeisi"Asanjarani, Azam"https://zbmath.org/authors/?q=ai:asanjarani.azamIn this paper, the rigidity of Finsler manifolds is studied by extending Obata's theorem and transnormal functions from Riemannian manifolds to Finsler manifolds.
Obata's theorem states that a Riemannian manifold which admits a nontrivial solution to Obata's equation is isometric to a sphere. The authors show that a similar result follows for a Finsler manifold, i.e., the Finsler-extended Obata's equation:
\[
\nabla^{H} \nabla^{H} \rho + C^2 \rho g =0 \tag{1}
\]
admits a nontrivial global solution if and only if the Finsler manifold \((M,g)\) is isometric to a sphere of radius \(1/C\). Here, \(\nabla^{H}\) is the Cartan horizontal covariant derivative, and \(\rho\) is a scalar function on \(M\). On their way, they also discuss the relation between the equation (1) and critical points of \(\rho\), and the flag curvature of the Finsler manifold. Especially, they investigate transnormal functions and use them for classification of complete Finsler manifolds. They show that the number of critical points \((0, 1, 2)\) of the transnormal functions classifies the Finsler manifolds into three categories.
It is proved that the Finsler manifold is homeomorphic to a sphere when the transnormal function \(\rho:M \to [a,b]\) has no critical points in \((a, b)\).
Reviewer: Erico Tanaka (Kagoshima)Curvatures for unions of WDC setshttps://zbmath.org/1522.530642023-12-07T16:00:11.105023Z"Pokorný, Dušan"https://zbmath.org/authors/?q=ai:pokorny.dusanSummary: We prove the existence of the curvature measures for a class of \({\mathcal{U}}_{\mathrm{WDC}}\) sets, which is a direct generalisation of \({\mathcal{U}}_{\mathrm{P\! R}}\) sets studied by \textit{J. Rataj} and \textit{M. Zähle} [Ann. Global Anal. Geom. 20, No. 1, 1--21 (2001; Zbl 0997.53062)]. Moreover, we provide a simple characterisation of \({\mathcal{U}}_{\mathrm{WDC}}\) sets in \(\mathbb{R}^2\) and prove that in \(\mathbb{R}^2\), the class of \({\mathcal{U}}_{\mathrm{WDC}}\) sets contains essentially all classes of sets known to admit curvature measures.
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to University College London under an exclusive licence. \textit{Mathematika} is published by the London Mathematical Society on behalf of University College London.}The translation number and quasi-morphisms on groups of symplectomorphisms of the diskhttps://zbmath.org/1522.530652023-12-07T16:00:11.105023Z"Maruyama, Shuhei"https://zbmath.org/authors/?q=ai:maruyama.shuheiThis paper is concerned with the construction of homogenous quasi-morphisms on groups of symplectomorphisms of the disk.
Let \(D=\{ (x,y) \in \mathbb R^2\mid x^2 + y^2 \leq 1 \}\) be the unit disk in \(\mathbb R^2\) and \(\omega = dx \wedge dy \) be the standard symplectic form on \(D\). Let \(G= \mathrm{Symp}(D)\) be the group of symplectomorphisms of \(D\) (which may not be the identity on the boundary \(\partial D\)). In this paper the author first constructs a homogeneous quasi-morphism on \(G\), extending the Calabi invariant. Recall that the restriction homomorphism \(\rho: G \to \mathrm{Diff}_+ (S^1)\), where \(\mathrm{Diff}_+ (S^1)\) is the group of orientation-preserving diffeomorphisms of the unit circle \(S^1 = \partial D\), is surjective, see [\textit{T. Tsuboi}, Trans. Am. Math. Soc. 352, No. 2, 515--524 (2000; Zbl 0937.57023)]. Denote by \(G_{\mathrm{rel}}\) the kernel of \(\rho\). The Calabi invariant \(\mathrm{Cal}: G_{\mathrm{rel}} \to \mathbb R\) is defined by
\[
\mathrm{Cal}(h) = \int_D h^*\eta \wedge \eta,
\]
where \(\eta\) is a \(1\)-form satisfying \(d\eta = \omega\). It is well known that this invariant is a surjective homomorphism and it is independent of the choice of \(\eta\). The author defines the map \(\tau_\eta: G \to \mathbb R\) in the same way:
\[
\tau_\eta (g) = \int_D g^*\eta \wedge \eta.
\]
This map is not a homomorphism and does depend on \(\eta\). It turns out that it is a quasi-morphism, its homogenization \(\bar{\tau}\) does not depend on \(\eta\) and it is an extension of the Calabi invariant. Moreover, there is another extension of the Calabi invariant previously introduced by \textit{T. Tsuboi} [loc. cit.], which is a homomorphism to \(\mathbb R\) from the universal covering group of \(G\). One of the main results of this paper (Theorem 1.1) establishes a relation between the two extensions, involving the translation number introduced by \textit{H. Poincaré} [C. R. Acad. Sci., Paris 90, 673--675 (1880; JFM 12.0588.01)].
In a similar way, the author constructs a homogeneous quasi-morphism \(\bar{\sigma}\) on the subgroup \(G_o\) of \(G\), consisting of symplectomorphisms preserving the origin, which extends a version of the flux homomorphism. More precisely, let \(G_{o,\mathrm{rel}} = G_{\mathrm{rel}} \cap G_o\). The flux homomorphism \(\mathrm{Flux}_{\mathbb R}\) is defined by
\[
\mathrm{Flux}_{\mathbb R}(h) = \int_\gamma h^*\eta - \eta,
\]
where \(\gamma\) is a path from the origin \(o\) to a point in the boundary \(\partial D\). This homomorphism is surjective and it is independent of the choice of \(\eta\) and \(\gamma\). As in the previous case, the quasi-morphism \(\bar{\sigma}\) relates with another extension of the flux homomorphism through the translation number (Theorem 1.2).
Finally, in the last section, the author shows that the difference \(\bar{\tau} -\pi \bar{\sigma}: G_{o} \to \mathbb R\) is a continuous homomorphism, extending the difference \(\mathrm{Cal} -\pi \mathrm{Flux}_{\mathbb R}\), although \(\mathrm{Cal}\) and \(\mathrm{Flux}_{\mathbb R}\) cannot can be extended to homomorphisms on \(G_o\).
Reviewer: Sílvia Anjos (Lisboa)A Bangert-Hingston theorem for starshaped hypersurfaceshttps://zbmath.org/1522.530662023-12-07T16:00:11.105023Z"Pellegrini, Alessio"https://zbmath.org/authors/?q=ai:pellegrini.alessioThe aim of this paper is to prove a version of the theorem of \textit{V. Bangert} and \textit{N. Hingston} [J. Differ. Geom. 19, 277--282 (1984; Zbl 0545.53036)] for starshaped hypersurfaces. Let \(Q\) be a closed manifold with non-trivial first Betti number that admits a non-trivial \(S^1\)-action, and \(\Sigma\subseteq T^*Q\) a nondegenerate starshaped hypersurface. In this paper, the author proves that the number of geometrically distinct Reeb orbits of period at most \(T\) on \(\Sigma\) grows at least logarithmically in \(T\).
The paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with Reeb orbits and starshaped domains. Section 3 is devoted to spectral invariants and minimax values. Here, the author introduces spectral invariants and compares them to minimax values on \(T^*Q\). Section 4 deals with pinching and Floer homologies. In this section, the author closely follows [\textit{L. Macarini} and \textit{F. Schlenk}, Math. Proc. Camb. Philos. Soc. 151, No. 1, 103--128 (2011; Zbl 1236.53063); \textit{M. Heistercamp}, ``The Weinstein conjecture with multiplicities on spherizations'', Preprint, \url{arXiv:1105.3886}; \textit{R. E. Wullschleger}, Counting Reeb chords on spherizations. Université de Neuchâtel (PhD thesis) (2014)] and constructs three sequences of non-degenerate Hamiltonians. Sections 5 and 6 are devoted to index relations and the main result respectively.
Reviewer: Ahmed Lesfari (El Jadida)Contact 3-manifolds with pseudo-parallel characteristic Jacobi operatorhttps://zbmath.org/1522.530672023-12-07T16:00:11.105023Z"Inoguchi, Jun-ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichi"Lee, Ji-Eun"https://zbmath.org/authors/?q=ai:lee.jieunSummary: In this article, we study contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator. Contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are \(M_\ell\)-manifolds (contact metric 3-manifolds with vanishing characteristic Jacobi operator) or generalized contact \((\kappa, \mu, \nu)\)-spaces. Moreover, we prove that contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are classified into four classes. In particular, we give a complete classification of homogeneous contact metric 3-manifolds with proper pseudo-parallel characteristic Jacobi operator.The geometry of some thermodynamic systemshttps://zbmath.org/1522.530682023-12-07T16:00:11.105023Z"Simoes, Alexandre Anahory"https://zbmath.org/authors/?q=ai:anahory-simoes.alexandre"Martín de Diego, David"https://zbmath.org/authors/?q=ai:martin-de-diego.david"Valcázar, Manuel Lainz"https://zbmath.org/authors/?q=ai:valcazar.manuel-lainz"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuelThis paper presents an alternative approach of the geometrical framework for studying some thermodynamic systems on a \((2n+1)\)-dimensional contact manifold, by using the so-called evolution vector field \(\mathcal{E}_H\) (described in terms of a skew-symmetric bracket of functions) instead of the contact vector field \(X_H\) used by other authors, both provided by a Hamiltonian \(H : T^{\ast}Q \times\mathbb{R}\to\mathbb{R}\), defined by \((q^i, p_i, S) \to H(q^i, p_i, S)\), where \((q^i)\) is the position, \((p_i)\) is the momentum and \(S\) is the entropy. Note that the integral curves of this evolution vector field describe the trajectories of a thermodynamic system which fulfills the first and second laws of thermodynamics, being thus, in the authors' opinion, a good candidate for the study of thermodynamic processes. Therefore, the authors investigate in their way the simple mechanical systems with friction (isolated systems) or composed thermodynamic systems without friction. Finally, the authors provide some geometric integrators for their used formalism.
For the entire collection see [Zbl 1468.68006].
Reviewer: Mircea Neagu (Braşov)Characterization of Whitney spheres among Lagrangian submanifolds with conformal Maslov formhttps://zbmath.org/1522.530692023-12-07T16:00:11.105023Z"Zhao, Entao"https://zbmath.org/authors/?q=ai:zhao.en-tao"Cao, Shunjuan"https://zbmath.org/authors/?q=ai:cao.shunjuanWhitney spheres are typical Lagrangian submanifolds in complex space forms. A Lagrangian submanifold \(M\) in a Kähler manifold is said to be a Lagrangian submanifold with conformal Maslov form if \(JH\) is a conformal vector field of \(M\). In this paper, the authors give new characterizations of Whitney spheres in complex space forms by the conformal Maslov form.
Let \(\mathbf{N}^n(4c)\) be a complex space form of complex dimension \(n\) with constant holomorphic sectional curvature \(4c\) with \(c \in {0,\pm 1}\). Let \(\phi : M^n \longrightarrow \mathbf{N}^n(4c)\) be an isometric immersion from an \(n\)-dimensional manifold \(M\) into \(\mathbf{N}^n(4c)\). Let \(H\) be the mean curvature vector. Let \(h(X,Y)\) be the second fundamental form. Define a modified second fundamental form \(B\) of \(M\) in \(\mathbf{N}^n(4c)\) by
\[
B(X,Y)=h(X,Y) - \frac{n}{n+2}(\langle X,Y\rangle H+\langle JX,H\rangle JY +\langle JY,H\rangle JX)
\]
for vector fields \(X,Y\) on \(M\). Let \(\gamma (n, |H|, c)\) denote the square of the nonnegative root of the equation
\[
\frac{3}{2} x^2 + \frac{n-2}{\sqrt{n(n-1)}} |H| x - \Big( (n+1) c + \frac{n^2}{n+2} |H|^2\Big) =0.
\]
The authors mainly prove two theorems in this paper:
Theorem. Let \(M\) be an \(n\)-dimensional compact Lagrangian submanifold with conformal Maslov form in \(\mathbf{N}^n(4c)\). Suppose \(|H|^2 + (n+1)(n+2)c /n^2 > 0\) for \(c<0\). If \(|B|\leq \gamma(n, |H|, c)\), then \(M\) is one of the following:
(i) A totally geodesic submanifold in \(\mathbb{CP}^n\);
(ii) A Whitney sphere in \(\mathbf{N}^n(4c)\);
(iii) A Clifford torus in \(\mathbb{CP}^2\).
Theorem. Let \(M\) be an \(n\)-dimensional non-minimal compact Lagrangian submanifold with conformal Maslov form in \(\mathbf{N}^n(4c)\).
(i) If \(c = 1\) and \(\int_M |B|^n d\mu < C_1(n)\), where \(C_1(n)\) is a positive constant depending only on \(n\), then \(M\) is either a totally geodesic submanifold or the Whitney sphere in \(\mathbb{CP}^n\):
(ii) If \(c = 0\) and \(\int_M B|^n d\mu < C_2(n)\), where \(C_2(n)\) is a positive constant depending only on \(n\), then \(M\) is the Whitney sphere in \(\mathbb{C}^n\);
(iii) If \(c = -1\), \(|H|^2- (n+1)(n+2)/n^2 \geq \tau\) for some \(\tau >0\) and \(\int _M |B|^n d\mu <C(n,\tau)\), where \(C(n,\tau)\) is a positive constant depending only on \(n\) and \(\tau\), then \(M\) is the Whitney sphere in \(\mathbb{CH}^n\).
Reviewer: Shiquan Ren (Singapore)Invariant almost contact structures and connections on the Lobachevsky spacehttps://zbmath.org/1522.530702023-12-07T16:00:11.105023Z"Rastrepina, A. O."https://zbmath.org/authors/?q=ai:rastrepina.anastasia-o"Surina, O. P."https://zbmath.org/authors/?q=ai:surina.olga-petrovnaAuthors' abstract: In this paper, we study the existence of invariant almost contact metric structures and connections in the Lobachevsky space using the Poincaré model and the group model of the Lobachevsky space. It is established that in the Lobachevsky space there exist left-invariant almost contact structures, among which an integrable normal almost contact metric structure is distinguished. All left-invariant linear connections consistent with the given structure are found, among which connections with zero curvature tensor are distinguished. It is proved that these connections are compatible with a foliation of an integrable normal almost contact metric structure in the sense that a unique geodesic belonging to the given foliation passes through each point in each direction tangent to this foliation. In addition to the Levi-Civita connections, in the Lobachevsky space there is also a metric connection with skew-symmetric torsion invariant under the full six-dimensional group of motions, as well as the only semisymmetric contact metric connection invariant under the four-dimensional subgroup of the group of motions.
Reviewer: Mohammed Guediri (Riyadh)Hessenberg varieties and Poisson sliceshttps://zbmath.org/1522.530712023-12-07T16:00:11.105023Z"Crooks, Peter"https://zbmath.org/authors/?q=ai:crooks.peter"Röser, Markus"https://zbmath.org/authors/?q=ai:roser.markusSummary: This expository article considers a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian \(G\)-variety \(\mu:G\times\mathcal{S}\longrightarrow\mathfrak{g}\) to each complex semisimple Lie algebra \(\mathfrak{g}\) with adjoint group \(G\) and fixed Kostant section \(\mathcal{S}\subseteq\mathfrak{g}\). This variety is one of Bielawski's hyperkähler slices, and it is central to \textit{G. W. Moore} and \textit{Y. Tachikawa}'s work [Proc. Sympos. Pure Math., 85, Am. Math. Soc., 191--207 (2012)] on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian \(G\)-varieties \(\overline{\mu}_{\mathcal{S}}:\overline{G\times\mathcal{S}}\longrightarrow\mathfrak{g}\) and \(\nu:\mathrm{Hess}\longrightarrow\mathfrak{g}\). The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification \(\overline{G}\), while the latter is the standard family of Hessenberg varieties. Each of \(\overline{\mu}\) and \(\nu\) is known to be a fibrewise compactification of \(\mu\).
We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our work is shown to be compatible with a Poisson isomorphism obtained by \textit{A. Bălibanu} [Represent. Theory 21, 132--150 (2017; Zbl 1428.20040)].
For the entire collection see [Zbl 1522.14005].Cosymplectic groupoidshttps://zbmath.org/1522.530722023-12-07T16:00:11.105023Z"Fernandes, Rui Loja"https://zbmath.org/authors/?q=ai:fernandes.rui-loja"Iglesias Ponte, David"https://zbmath.org/authors/?q=ai:iglesias-ponte.davidSummary: A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic structure. We provide several structural results for cosymplectic groupoids and we discuss the relationship between cosymplectic groupoids, Poisson groupoids of corank 1, and oversymplectic groupoids of corank 1.On symmetries of singular foliationshttps://zbmath.org/1522.530732023-12-07T16:00:11.105023Z"Louis, Ruben"https://zbmath.org/authors/?q=ai:louis.rubenIn this interesting paper the author shows that a weak symmetry action of a Lie algebra \(g\) on a singular foliation \(F\) induces a unique up to homotopy Lie \(\infty\)-morphism from \(g\) to the DGLA of vector fields on a universal Lie \(\infty\)-algebroid of \(F\). Such a morphism was called an \(L_\infty\)-algebra action in [\textit{R. A. Mehta} and \textit{M. Zambon}, Differ. Geom. Appl. 30, No. 6, 576--587 (2012; Zbl 1267.58003)]. From this general result several geometrical consequences are deduced. It is given an example of a Lie algebra action on an affine subvariety which cannot be extended to the ambient space. Finally, the notion of bi-submersion towers over a singular foliation and lift symmetries to those is introduced.
Reviewer: Liviu Popescu (Craiova)A Poisson bracket on the space of Poisson structureshttps://zbmath.org/1522.530742023-12-07T16:00:11.105023Z"Machon, Thomas"https://zbmath.org/authors/?q=ai:machon.thomasLet \(M\) be a smooth, closed and orientable manifold. The author considers the set of all Poisson structures on \(M\), denoted \(\mathcal{P}(M)\) and shows that \(\mathcal{P}(M)\) has itself a family of Poisson structures \(\{\,,\}_{\mu}\), depending on a choice of a volume form \(\mu\). The motivation for this work comes from ideal fluid dynamics. The aim is to extend the Poisson bracket on the space of Poisson structures and to define a Poisson bracket on the space of admissible functions. The bracket is explicitly given and the corresponding proofs are detailed. The author considers also the space of symplectic manifolds with a symplectic volume form. In this case, he constructs a further and related Poisson bracket. He studies the induced flow and gives a description in terms of the symplectic cohomology groups introduced by \textit{L.-S. Tseng} and \textit{S.-T. Yau} [J. Differ. Geom. 91, No. 3, 383--416 (2012; Zbl 1275.53079)].
Reviewer: Angela Gammella-Mathieu (Metz)Atiyah and Todd classes of regular Lie algebroidshttps://zbmath.org/1522.530752023-12-07T16:00:11.105023Z"Xiang, Maosong"https://zbmath.org/authors/?q=ai:xiang.maosongA dg manifold (or a Q-manifold) is a \(\mathbb{Z}\)-graded smooth manifold equipped with a homological vector field Q, i.e., a degree \(+1\) derivation of square zero on the algebra of smooth functions.
A Lie algebroid \((A, \rho_A, [-,-]_A)\) over a smooth manifold \(M\) is said to be {\em regular} if its anchor \(\rho_A\) is of constant rank. The kernel \(K = \ker(\rho_A)\) together with the restriction \([-,-]_K\) of the Lie bracket \([-,-]_A\) onto \(\Gamma (K)\) is a bundle of Lie algebras; the image \(F = \operatorname{Im}(\rho_A) \subseteq TM\) as the tangent bundle of the regular characteristic foliation, is a Lie subalgebroid of the tangent Lie algebroid \(TM\). In other words there is a short exact sequence
\[
0\rightarrow K \stackrel{i} \rightarrow A \stackrel{\rho_A} \rightarrow F\rightarrow 0
\]
of Lie algebroids over \(M\), known as the Atiyah sequence of \(A\).
In this paper, the author studies the Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids and proves that these classes fit into a short exact sequence called Atiyah sequence.
Reviewer: Osman Mucuk (Kayseri)A non commutative Kähler structure on the Poincaré disk of a \(C^*\)-algebrahttps://zbmath.org/1522.530762023-12-07T16:00:11.105023Z"Andruchow, Esteban"https://zbmath.org/authors/?q=ai:andruchow.esteban"Corach, Gustavo"https://zbmath.org/authors/?q=ai:corach.gustavo"Recht, Lázaro"https://zbmath.org/authors/?q=ai:recht.lazaro-aThe article is the third in a series of articles by the authors, where they study the Poincaré disc of a unital \(C^{\ast}\)-algebra. Here they endow the Poincaré disc as such with a noncommutative homogeneous Kähler structure and study the associated moment map. The principal result is the generalization of the classical Atiyah-Guillemin-Sternberg theorem about the convexity of the image of this moment map, in the presence of a trace. To this end, the authors carry out an appropriate geometric (pre)quantization program.
For the entire collection see [Zbl 1492.47001].
Reviewer: Iakovos Androulidakis (Athína)Towers of Looijenga pairs and asymptotics of ECH capacitieshttps://zbmath.org/1522.530772023-12-07T16:00:11.105023Z"Wormleighton, Ben"https://zbmath.org/authors/?q=ai:wormleighton.benSummary: ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are \(O(1)\). We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising results of Hutchings and of the author, and derive new obstructions to embeddings between toric domains of the same volume. We also propose two invariants to more precisely describe when convergence occurs in the toric case. Our methods are largely non-toric in nature, and apply more widely to towers of polarised Looijenga pairs.A short proof of cuplength estimates on Lagrangian intersectionshttps://zbmath.org/1522.530782023-12-07T16:00:11.105023Z"Gong, Wenmin"https://zbmath.org/authors/?q=ai:gong.wenminLet \(M\) be a closed \(n\)-manifold with the canonical symplectic form \(\omega\) on the cotangent bundle \(T^*M\). Given \(H\in C^\infty([0,1]\times T^*M)\), the Hamiltonian vector field \(X_H\) is determined by the equation \(dH=-\omega(X_H,.)\). This vector field gives rise to the flow \(\varphi^t\) and its time-one map \(\varphi\). If \(H\) is asymptotically constant then \(\varphi\) is a Hamiltonian diffeomorphism with compact support. Let \(O_M\) denote the zero section of \(T^*M\). The author reproves the following cup-length estimate:
\[
\sharp \varphi (O_m)\cap O_M \geq cl(M).
\]
The method is making use of the Lagrangian properties of spectral invariants from Floer theory (see [\textit{A. Floer}, Commun. Pure Appl. Math. 42, No. 4, 335--356 (1989; Zbl 0683.58017); \textit{Y.-G. Oh}, J. Differ. Geom. 46, No. 3, 499--577 (1997; Zbl 0926.53031); Commun. Anal. Geom. 7, No. 1, 1--55 (1999; Zbl 0966.53055)]).
Reviewer: Zdzisław Dzedzej (Gdańsk)Lagrangian fields, Calabi functions, and local symplectic groupoidshttps://zbmath.org/1522.530792023-12-07T16:00:11.105023Z"Karabegov, Alexander"https://zbmath.org/authors/?q=ai:karabegov.alexander-vSummary: A Lagrangian field on a symplectic manifold \(M\) is a family \({\Lambda} = \{ {\Lambda}_x | x \in M \}\) of pointed Lagrangian submanifolds of \(M\). This notion is a generalization of a real Lagrangian polarization for which each \({\Lambda}_x\) is the leaf containing \(x\). Two Lagrangian fields \(\Lambda\) and \(\widetilde{{\Lambda}}\) are called transversal if \({\Lambda}_x\) intersects \(\widetilde{{\Lambda}}_x\) transversally at \(x\) for every \(x \in M\). Two transversal Lagrangian fields determine an almost para-Kähler structure on \(M\). We construct a local symplectic groupoid on a neighborhood of the zero section of \(T^\ast M\) from two transversal Lagrangian fields on \(M\). The Lagrangian manifold of \(n\)-cycles of this groupoid in \((T^\ast M)^n\) has a generating function whose germ around the diagonal of \(M^n\) is given by the \(n\)-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of \(M^2\) obtained from the symplectic form on \(M\).The strong homotopy structure of BRST reductionhttps://zbmath.org/1522.530802023-12-07T16:00:11.105023Z"Esposito, Chiara"https://zbmath.org/authors/?q=ai:esposito.chiara"Kraft, Andreas"https://zbmath.org/authors/?q=ai:kraft.andreas"Schnitzer, Jonas"https://zbmath.org/authors/?q=ai:schnitzer.jonasThis paper proposes a reduction scheme for equivariant polydifferential operators. Let us recall that in the theory of deformation quantization, the phase space is described by a Poisson manifold \(M\). Quantifying \(M\) refers to the construction of a star product on \(M\). Let us recall that the existence and the classification of star products in the setting of general Poisson manifolds are obtained by Kontsevich's formality theorem. The motivation of this paper is to investigate the compatibility of deformation and phase space reduction in the Poisson setting. The authors obtain the desired reduction \(L_{\infty}\)-morphism by applying an explicit version of the homotopy transfer theorem. Finally, the authors prove that the reduced star product induced by reduction coincides with the reduced star product obtained via the formal Koszul complex.
Reviewer: Angela Gammella-Mathieu (Metz)Quantization of restricted Lagrangian subvarieties in positive characteristichttps://zbmath.org/1522.530812023-12-07T16:00:11.105023Z"Mundinger, Joshua"https://zbmath.org/authors/?q=ai:mundinger.joshuaSummary: \textit{R. Bezrukavnikov} and \textit{D. Kaledin} [J. Am. Math. Soc. 21, No. 2, 409--438 (2008; Zbl 1138.53067)] introduced quantizations of symplectic varieties \(X\) in positive characteristic which endow the Poisson bracket on \(X\) with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties \(Y\) of \(X\) to modules over such quantizations. If the ideal sheaf of \(Y\) is a restricted Lie subalgebra of the structure sheaf of \(X\), we show that there is a certain cohomology class which vanishes if and only if a line bundle on \(Y\) admits a quantization.Pinched hypersurfaces are compacthttps://zbmath.org/1522.530822023-12-07T16:00:11.105023Z"Bourni, Theodora"https://zbmath.org/authors/?q=ai:bourni.theodora"Langford, Mat"https://zbmath.org/authors/?q=ai:langford.mat"Lynch, Stephen"https://zbmath.org/authors/?q=ai:lynch.stephenA convex hypersurface \(M^n, \ n\geq 2,\) in \({\mathbb R}^{n+1}\) is said to be \(\alpha\)-pinched for some \(\alpha>0\) if it satisfies \(k_1\geq \alpha k_n\) on \(M,\) where \(k_1\leq k_2\leq \cdots \leq k_n\) denote the principal curvatures of \(M.\) A result of \textit{R. S. Hamilton} [Commun. Anal. Geom. 2, No. 1, 167--172 (1994; Zbl 0843.53002)] asserts that a strictly convex complete hypersurface in \({\mathbb R}^{n+1}\) that bounds a region and which is \(\alpha\)-pinched must be compact. His proof was based on the observation that the Gauss map of such a hypersurface must be quasi-conformal.
The authors prove the following version of Hamilton's result: Assume that \(M^n,\ n\geq 2,\) is a convex hypersurface in \({\mathbb R}^{n+1}\) with bounded curvature. If \(M\) is \(\alpha\)-pinched, then \(M\) is either a hyperplane or it is compact. This is achieved by evolving the hypersurface under the mean curvature flow and using a blow up argument.
Reviewer: H. A. Gururaja (Tirupati)Bochner formulas, functional inequalities and generalized Ricci flowhttps://zbmath.org/1522.530832023-12-07T16:00:11.105023Z"Kopfer, Eva"https://zbmath.org/authors/?q=ai:kopfer.eva"Streets, Jeffrey"https://zbmath.org/authors/?q=ai:streets.jeffrey-dThe authors consider the generalized Ricci flow, which is a system of two equations in terms of a \(t\)-family of Riemannian metrics and a \(t\)-family of \(2\)-forms. Solutions to generalized Ricci flow are super-solutions of the Ricci flow. The results are summarized in two theorems.
Theorem 1.1 is the universal Poincaré inequality and the log-Sobolev inequality along the flow. This applies the Bochner formula for the Bismut connection, and generalizes the work of \textit{H.-J. Hein} and \textit{A. Naber} [Commun. Pure Appl. Math. 67, No. 9, 1543--1561 (2014; Zbl 1297.53046)] on the Ricci flow. It is possible to use these inequalities to characterize super-solutions to generalized Ricci flow.
Theorem 1.2 extends Theorem 1.1 to the path space. It consists of (equivalent) characterizations of generalized Ricci flow by universal Poincaré and log-Sobolev inequalities on path space, for the associated Malliavin gradient and the Ornstein-Uhlenbeck operator. Along the way, using the 2-form potential, the authors define a twisted connection on the space-time. With the help of an anti-development map, this yields the Brownian motion on the frame bundle. Therefore, they obtain a parallel gradient for martingales, and the Malliavin gradient on path space. The Ornstein-Uhlenbeck operator is determined by the adapted geometry on path space, and is the composition of the Malliavin gradient and its adjoint. The parallel gradient is the time derivative of the Malliavin gradient. Moreover, they obtain Bochner formula for the Malliavin gradient/parallel gradient, which generalizes the construction along Ricci flow by \textit{M. Arnaudon} et al. [C. R., Math., Acad. Sci. Paris 346, No. 13--14, 773--778 (2008; Zbl 1144.58019)], \textit{R. Haslhofer} and \textit{A. Naber} [J. Eur. Math. Soc. (JEMS) 20, No. 5, 1269--1302 (2018; Zbl 1397.53051)], and \textit{C. Kennedy} [``A Bochner formula on path space for the Ricci flow'', Preprint, \url{arXiv:1909.04193}]. These Bochner formulas on path spaces imply Theorem 1.2. When the equivalent conditions of generalized Ricci flow solutions are satisfied, the authors also prove a Poincaré Hessian estimate and a log-Sobolev Hessian estimate along the flow.
Reviewer: Yuanqi Wang (Stony Brook)Hyper-elastic Ricci flow: gradient flow, local existence-uniqueness, and a Perelman energy functionalhttps://zbmath.org/1522.530842023-12-07T16:00:11.105023Z"Slemrod, Marshall"https://zbmath.org/authors/?q=ai:slemrod.marshallSummary: The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the a free energy. In this paper hyper-elastic Ricci flow is shown to possess three properties derived by G. Perelman for classical Ricci flow, specifically it is diffeomorphically equivalent to a gradient flow, unique smooth solutions exist locally in time, and the system possesses a non-decreasing energy function.Geometric flow, multiplier ideal sheaves and optimal destabilizer for a Fano manifoldhttps://zbmath.org/1522.530852023-12-07T16:00:11.105023Z"Hisamoto, Tomoyuki"https://zbmath.org/authors/?q=ai:hisamoto.tomoyukiUsing the Ricci curvature formalism, the author answers a question by \textit{S. K. Donaldson} [J. Differ. Geom. 70, No. 3, 453--472 (2005; Zbl 1149.53042)] to understand whether the lower bound of the Calabi functional is achieved by a sequence of normalized Donaldson-Futaki invariants.
Reviewer: Marek Jarnicki (Kraków)Kähler-Ricci flow and conformal submersionhttps://zbmath.org/1522.530862023-12-07T16:00:11.105023Z"Hoan, Nguyen The"https://zbmath.org/authors/?q=ai:hoan.nguyen-theSummary: We study singularity formation of Kähler-Ricci flow on a Kähler manifold that admits a horizontally homothetic conformal submersion into another Kähler manifold. We will derive necessary and sufficient conditions for the preservation of horizontally homothetic conformal submersion along the flow and establish the formation of type I singularity together with a standard splitting of the Cheeger-Gromov limit. This generalizes the setup of Calabi symmetry that was discussed in [\textit{F. T. H. Fong}, Trans. Am. Math. Soc. 366, No. 2, 563--589 (2014; Zbl 1294.53061); \textit{J. Song} and \textit{B. Weinkove}, J. Reine Angew. Math. 659, 141--168 (2011; Zbl 1252.53080)] and produces novel proofs for the established results.On an elastic flow for parametrized curves in \(\mathbb{R}^n\) suitable for numerical purposeshttps://zbmath.org/1522.530872023-12-07T16:00:11.105023Z"Pozzi, Paola"https://zbmath.org/authors/?q=ai:pozzi.paolaOn the space of closed, regular, smooth curves in \(\mathbb{R}^n\) consider an energy functional consisting of the sum of the total squared curvature functional (bending energy) and a penalization term proportional to the Dirichlet energy. The author proves that short-time existence of solutions for the negative gradient flow of this energy implies global existence. The motivation for considering this flow comes from the fact that it has good numerical properties. The paper is clear and well written.
Reviewer: Florin Catrina (New York)Real hypersurfaces with pseudo-Ricci-Bourguignon soliton in the complex two-plane Grassmannianshttps://zbmath.org/1522.530882023-12-07T16:00:11.105023Z"Suh, Young Jin"https://zbmath.org/authors/?q=ai:suh.young-jin"Woo, Changhwa"https://zbmath.org/authors/?q=ai:woo.changhwaSummary: In this paper, we have investigated a pseudo-Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian \(G_2(\mathbb{C}^{m+2})\). By using pseudo-anti commuting Ricci tensor, we give a complete classification of Hopf pseudo-Ricci-Bourguignon soliton real hypersurfaces in \(G_2( \mathbb{C}^{m+2})\). Moreover, we have proved that there exists a non-trivial classification of gradient pseudo-Ricci-Bourguignon soliton \((M, \xi, \eta, \Omega, \theta, \gamma, g)\) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian \(G_2( \mathbb{C}^{m + 2})\). In the class of contact hypersurface in \(G_2( \mathbb{C}^{m + 2})\), we prove that there does not exist a gradient pseudo-Ricci-Bourguignon soliton in \(G_2( \mathbb{C}^{m + 2})\).Complex structures, T-duality and worldsheet instantons in Born sigma modelshttps://zbmath.org/1522.530892023-12-07T16:00:11.105023Z"Kimura, Tetsuji"https://zbmath.org/authors/?q=ai:kimura.tetsuji"Sasaki, Shin"https://zbmath.org/authors/?q=ai:sasaki.shin"Shiozawa, Kenta"https://zbmath.org/authors/?q=ai:shiozawa.kentaSummary: We investigate doubled (generalized) complex structures in \(2D\)-dimensional Born geometries where T-duality symmetry is manifestly realized. We show that Kähler, hyperkähler, bi-hermitian and bi-hypercomplex structures of spacetime are implemented in Born geometries as doubled structures. We find that the Born structures and the generalized Kähler (hyperkähler) structures appear as subalgebras of bi-quaternions and split-tetra-quaternions. We find parts of these structures are classified by Clifford algebras. We then study the T-duality nature of the worldsheet instantons in Born sigma models. We show that the instantons in Kähler geometries are related to those in bi-hermitian geometries in a non-trivial way.On naturality of the Ozsváth-Szabó contact invarianthttps://zbmath.org/1522.570322023-12-07T16:00:11.105023Z"Hedden, Matthew"https://zbmath.org/authors/?q=ai:hedden.matthew"Tovstopyat-Nelip, Lev"https://zbmath.org/authors/?q=ai:tovstopyat-nelip.levSummary: We discuss functoriality properties of the Ozsváth-Szabó contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.
For the entire collection see [Zbl 1515.57005].Every real 3-manifold is real contacthttps://zbmath.org/1522.570432023-12-07T16:00:11.105023Z"Cengiz, Merve"https://zbmath.org/authors/?q=ai:cengiz.merve"Öztürk, Ferit"https://zbmath.org/authors/?q=ai:ozturk.feritThe authors re-prove several classical results from 3-dimensional contact topology in the category of real contact manifolds. A real contact 3-manifold is a closed manifold together with an orientation-preserving involution \(c_M\) such that the fixed point set of \(c_M\) is either empty or 1-dimensional, and a real contact structure, i.e., a contact structure such that
\begin{itemize}
\item \(c_M\) preserves the contact structure,
\item \(dc_M\) reverses the co-orientation of the contact structure.
\end{itemize}
The main result of the paper states that every real 3-manifold admits a real contact structure. The most important tools developed in this paper to prove this result are a \(c_M\)-equivariant version of contact Dehn-surgery and a Lickorish-Wallace theorem for real manifolds, stating that any closed orientable real 3-manifold can be obtained via equivariant Dehn-surgery along a certain link in the standard real \(S^3\). As an application it is shown that the tight contact structures on \(S^2\times S^1\) and on certain lens spaces are real.
Reviewer: Jakob Hedicke (Montréal)Distance and intersection number in the curve graph of a surfacehttps://zbmath.org/1522.570472023-12-07T16:00:11.105023Z"Birman, Joan S."https://zbmath.org/authors/?q=ai:birman.joan-s"Morse, Matthew J."https://zbmath.org/authors/?q=ai:morse.matthew-j"Wrinkle, Nancy C."https://zbmath.org/authors/?q=ai:wrinkle.nancy-cSummary: In this work, we study the cellular decomposition of \(S\) induced by a filling pair of curves \(v\) and \(w, Dec_{v,w}(S)=S\setminus (v \cup w)\), and its connection to the distance function \(d(v, w)\) in the curve graph of a closed orientable surface \(S\) of genus \(g\). Building on the work of Leasure, efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in [\textit{J. Birman} et al., Math. Ann. 366, No. 3--4, 1253--1279 (2016; Zbl 1350.05022)], giving an algorithm that begins with a pair of non-separating filling curves that determine vertices \((v,w)\) in the curve graph of a closed orientable surface \(S\) and computing from them a finite set of \textit{efficient} geodesics. We extend the tools of efficient geodesics to study the relationship between distance \(d(v,w)\), intersection number \(i(v,w)\), and \(Dec_{v,w}(S)\). The main result is the development and analysis of particular configurations of rectangles in \(Dec_{v,w}(S)\) called \textit{spirals}. We are able to show that, with appropriate restrictions, the efficient geodesic algorithm can be used to build an algorithm that reduces \(i(v,w)\) while preserving \(d(v,w)\). At the end of the paper, we note a connection between our work and the notion of extending geodesics.The dilogarithm and abelian Chern-Simonshttps://zbmath.org/1522.570602023-12-07T16:00:11.105023Z"Freed, Daniel S."https://zbmath.org/authors/?q=ai:freed.daniel-s"Neitzke, Andrew"https://zbmath.org/authors/?q=ai:neitzke.andrewThe \textit{dilogarithm function}, in its simplest form \(\mathrm{Li}_2(z)=\sum_{n=1}^\infty z^n/n^2\), which is defined in \(\{z\in\mathbb{C}\colon |z|<1\}\) and analytically continued to \(\mathbb{C}\setminus[1,\infty)\), has a long history. During the recent decades, the dilogarithm has been playing a significant role in hyperbolic geometry, algebraic \(K\)-theory, conformal field theory, and beyond. As an important variant, the \textit{enhanced Rogers dilogarithm}
\begin{align*}
L&:\{(u_1,u_2)\in\mathbb{C}^2\colon e^{u_1}+e^{u_2}=1\}\to\mathbb{C}/4\pi^2\mathbb{Z}, \\
z&\mapsto \mathrm{Li}_2(z)+\frac{1}{2}\log(z)\log(1-z)
\end{align*}
(which is well-defined) is shown in [\textit{J. L. Dupont}, J. Pure Appl. Algebra 44, 137--164 (1987; Zbl 0624.57024)] to be closely related to a Chern-Cheeger-Simons characteristic class of flat \(\mathrm{SL}(2,\mathbb{C})\)-bundles.
In the present paper, \(L\) is constructed using spin \textit{Chern-Simons theory} with gauge group \(\mathbb{C}^\times\), which is a symmetric monoidal functor \[\mathrm{Bord}_{\langle 2,3\rangle}(\mathrm{Spin}_3\times(\mathbb{C}^\times)^\nabla)\to s\mathrm{Line}_{\mathbb{C}}.\] Here the domain is the category whose objects are closed spin \(2\)-manifolds \(Y\) equipped with a \(\mathbb{C}^\times\)-connection \(\Theta_Y\), and a morphism \((Y_0,\Theta_0)\to(Y_1,\Theta_1)\) is a compact spin \(3\)-manifold equipped with a \(\mathbb{C}^\times\)-connection; the codomain \(s\mathrm{Line}_{\mathbb{C}}\) is the Picard groupoid of \(\mathbb{Z}/2\mathbb{Z}\)-graded super lines.
Subsequently, geometric proofs are given for some standard properties of \(L\), namely, the transformation law under a canonical \(\mathbb{Z}^2\)-action, the reflection identity, and the \(5\)-term relation. These proofs are impressively interesting.
Reviewer: Haimiao Chen (Beijing)Singularity classes of special multi-flags. Ihttps://zbmath.org/1522.580022023-12-07T16:00:11.105023Z"Mormul, Piotr"https://zbmath.org/authors/?q=ai:mormul.piotrIn this paper, the author investigates on singularity classes of special multi-flags. More precisely, the paper contributes on the study regarding the construction of the geometric singularity classes of special multi-flags.
Reviewer: Savin Treanţă (Bucureşti)On differential operators generated by geometric structureshttps://zbmath.org/1522.580052023-12-07T16:00:11.105023Z"Tudoran, Răzvan M."https://zbmath.org/authors/?q=ai:tudoran.razvan-micuThe author introduces and studies two classes of differential operators generated by geometric structures (that is, non-degenerate \((0,2)\)-tensor fields) on general connected, Hausdorff, paracompact and smooth manifolds. The first class of operators extends the concepts of gradient vector field from Riemannian or semi-Riemannian geometry, and Hamiltonian vector field from symplectic geometry. The second class of operators generalizes the Laplace-Beltrami operator from Riemannian geometry, and the d'Alembert operator from Lorentzian geometry.
Reviewer: Rodica Luca (Iaşi)Introduction of \(T\)-harmonic mapshttps://zbmath.org/1522.580072023-12-07T16:00:11.105023Z"Aminian, Mehran"https://zbmath.org/authors/?q=ai:aminian.mehranSummary: In this paper, we introduce a second order linear differential operator \(\overset{T}{\square}:C^{\infty} (M) \rightarrow C^{\infty} (M)\) as a natural generalization of Cheng-Yau operator, [\textit{S.-Y. Cheng} and \textit{S.-T. Yau}, Math. Ann. 225, 195--204 (1977; Zbl 0349.53041)], where \(T\) is a \((1, 1)\)-tensor on Riemannian manifold \((M, h)\), and then we show on compact Riemannian manifolds, \(\mathrm{div}T = \mathrm{div}T^t\), and if \(\mathrm{div}T = 0\), and \(f\) be a smooth function on \(M\), the condition \(\overset{T}{\square} f = 0\) implies that \(f\) is constant. Hereafter, we introduce \(T\)-energy functionals and by deriving variations of these functionals, we define \(T\)-harmonic maps between Riemannian manifolds, which is a generalization of \(L_k\)-harmonic maps introduced in [\textit{M. Aminian} and \textit{S. M. B. Kashani}, Acta Math. Vietnam. 42, No. 3, 471--490 (2017; Zbl 1373.53072)]. Also we have studied \(fT\)-harmonic maps for conformal immersions and as application of it, we consider \(fL_k\)-harmonic hypersurfaces in space forms, and after that we classify complete \(fL_1\)-harmonic surfaces, some \(fL_k\)-harmonic isoparametric hypersurfaces, \(fL_k\)-harmonic weakly convex hypersurfaces, and we show that there exists no compact \(fL_k\)-harmonic hypersurface either in the Euclidean space or in the hyperbolic space or in the Euclidean hemisphere. As well, some properties and examples of these definitions are given.On the normal stability of triharmonic hypersurfaces in space formshttps://zbmath.org/1522.580082023-12-07T16:00:11.105023Z"Branding, Volker"https://zbmath.org/authors/?q=ai:branding.volkerSummary: This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere \(\phi :\mathbb{S}^m (1/\sqrt{3})\hookrightarrow \mathbb{S}^{m+1}\) is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.Computing harmonic maps between Riemannian manifoldshttps://zbmath.org/1522.580092023-12-07T16:00:11.105023Z"Gaster, Jonah"https://zbmath.org/authors/?q=ai:gaster.jonah"Loustau, Brice"https://zbmath.org/authors/?q=ai:loustau.brice"Monsaingeon, Léonard"https://zbmath.org/authors/?q=ai:monsaingeon.leonardIn a previous paper [\textit{J. Gaster} et al., ``Computing discrete equivariant harmonic maps'', Preprint, \url{arXiv:1810.11932}], the authors have shown that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper they study the convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. They present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and offer a systematic method to construct such weighted triangulations in the two-dimensional case. Their computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.
Reviewer: Vladimir Balan (Bucureşti)Best constant and mountain-pass solutions for a supercritical Hardy-Sobolev problem in the presence of symmetrieshttps://zbmath.org/1522.580112023-12-07T16:00:11.105023Z"Mesmar, Hussein"https://zbmath.org/authors/?q=ai:mesmar.husseinLet \(G\) be a group of isometries of a Riemannian manifold. The author looks for the existence of \(G\)-invariant positive solutions of a suitable nonlinear equation. Concerning the proof, he uses the Aubin minimization and the Mountain-Pass lemma of Ambrosetti-Rabinowitz. Also, he finds the best constant in the associated Hardy-Sobolev inequality.
Reviewer: Mohammed El Aïdi (Bogotá)Time analyticity for the parabolic type Schrödinger equation on Riemannian manifold with integral Ricci curvature conditionhttps://zbmath.org/1522.580142023-12-07T16:00:11.105023Z"Wang, Wen"https://zbmath.org/authors/?q=ai:wang.wen|wang.wen.2On a complete Riemannian manifold with integral Ricci curvature condition, the author states that a smooth solution of exponential growth of a suitable parabolic Schrödinger equation is analytic in the Cartesian product of a geodesic ball with a unit interval (Theorem 2.1).
Reviewer: Mohammed El Aïdi (Bogotá)On the spectrum of certain Hadamard manifoldshttps://zbmath.org/1522.580162023-12-07T16:00:11.105023Z"Ballmann, Werner"https://zbmath.org/authors/?q=ai:ballmann.werner"Mukherjee, Mayukh"https://zbmath.org/authors/?q=ai:mukherjee.mayukh"Polymerakis, Panagiotis"https://zbmath.org/authors/?q=ai:polymerakis.panagiotisThe principal result states that the spectrum of \(M\), a homogeneous Hadamard manifold, is the interval \([\frac{h^2}{4},\infty)\) such that \(h\) is the mean curvature of \(N\), the connected Lie subgroup of \(A\ltimes N\cong M\) such that \(A\) is an abelian group.
Reviewer: Mohammed El Aïdi (Bogotá)Bootstrap bounds on closed hyperbolic manifoldshttps://zbmath.org/1522.580172023-12-07T16:00:11.105023Z"Bonifacio, James"https://zbmath.org/authors/?q=ai:bonifacio.jamesSummary: The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an upper bound on the gap between two consecutive nonzero eigenvalues of the Laplace-Beltrami operator in terms of the smaller eigenvalue, an upper bound on the smallest eigenvalue of the rough Laplacian on symmetric, transverse-traceless, rank-2 tensors, and bounds on integrals of products of eigenfunctions and eigentensors. Our strongest bounds involve numerically solving semidefinite programs and are presented as exclusion plots. We also prove the analytic bound \(\lambda_{i+1} \leq 1/2 + 3\lambda_i + \sqrt{\lambda_i^2+2\lambda_i+1/4}\) for consecutive nonzero eigenvalues of the Laplace-Beltrami operator on closed orientable hyperbolic surfaces. We give examples of genus-2 surfaces that nearly saturate some of these bounds. To derive the consistency conditions, we make use of a transverse-traceless decomposition for symmetric tensors of arbitrary rank.Symmetry of solutions to semilinear PDEs on Riemannian domainshttps://zbmath.org/1522.580202023-12-07T16:00:11.105023Z"Bisterzo, Andrea"https://zbmath.org/authors/?q=ai:bisterzo.andrea"Pigola, Stefano"https://zbmath.org/authors/?q=ai:pigola.stefanoThe paper is concerned with symmetry properties for solutions to the Dirichlet problem for semilinear PDEs on Riemannian domains. The authors introduce a general framework where the symmetry issue can be formulated, and they provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the framework from the very beginning.
Reviewer: Simone Secchi (Milano)Systoles and Lagrangians of random complex algebraic hypersurfaceshttps://zbmath.org/1522.600212023-12-07T16:00:11.105023Z"Gayet, Damien"https://zbmath.org/authors/?q=ai:gayet.damienAs it is well known, the systole \(\mathrm{Sys}(M,g)\) of a complete Riemannian manifold \((M,g)\) is the infimum of the length of non-contractible loops in \((M,g)\). When \(M\) is a closed manifold, the systole is realized by the shortest non-contractible geodesic loop. The classical isosystolic inequalities take the form \(\mathrm{Vol}(M,g)\geq k (\mathrm{Sys}(M,g))^{n}\), for any metric \(g\) on the \(n\)-dimensional manifold \(M\). The first of such type of inequalities was obtained by Loewner in an unpublished paper written in 1949 where he proved that for any surface \(S\) of the topological type of a torus, \(\textit{Area}(S,g) \geq \frac{\sqrt{3}}{2} (\mathrm{Sys}(S,g))^{2}\).
One of the goals of the paper under review is to obtain a uniform positive lower bound for the probability that a projective complex curve in \(\mathbb{C}P^{2}\) of given degree equipped with the restriction of the ambient metric has a systole of small size, which is an analog of a similar bound for hyperbolic curves given by \textit{M. Mirzakhani} [J. Differ. Geom. 94, No. 2, 267--300 (2013; Zbl 1270.30014)]. In order to prove that, the author obtains some results about Lagrangian submanifolds on a smooth complex projective hypersurface in \(\mathbb{C}P^{n}\). Although these are deterministic results, they are obtained as consequences of a more precise probabilistic theorem proved in the present paper, which is inspired by a 2014 result by \textit{D. Gayet} and \textit{J.-Y. Welschinger} [J. Lond. Math. Soc., II. Ser. 90, No. 1, 105--120 (2014; Zbl 1326.14139)] on random real algebraic geometry, together with quantitative Moser-type constructions.
Reviewer: Fernando Etayo Gordejuela (Santander)Geometric rough paths on infinite dimensional spaceshttps://zbmath.org/1522.600782023-12-07T16:00:11.105023Z"Grong, Erlend"https://zbmath.org/authors/?q=ai:grong.erlend"Nilssen, Torstein"https://zbmath.org/authors/?q=ai:nilssen.torstein-k"Schmeding, Alexander"https://zbmath.org/authors/?q=ai:schmeding.alexanderSummary: Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For \(\alpha \in(1/3, 1/2)\), we give criteria for when we can approximate Banach space-valued weakly geometric \(\alpha \)-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.Discrete hyperbolic curvature flow in the planehttps://zbmath.org/1522.651392023-12-07T16:00:11.105023Z"Deckelnick, Klaus"https://zbmath.org/authors/?q=ai:deckelnick.klaus"Nürnberg, Robert"https://zbmath.org/authors/?q=ai:nurnberg.robertSummary: Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by \textit{M. E. Gurtin} and \textit{P. Podio-Guidugli} [SIAM J. Math. Anal. 22, No. 3, 575--586 (1991; Zbl 0762.35068)] to model certain wave phenomena in solid-liquid interfaces. We introduce a semidiscrete finite difference method for the approximation of hyperbolic curvature flow and prove error bounds for natural discrete norms. We also present numerical simulations, including the onset of singularities starting from smooth strictly convex initial data.Visualizing Thurston's geometries. Paper from the 33rd Brazilian mathematics colloquium -- 33\degree Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, Brazil, August 2--6, 2021https://zbmath.org/1522.680072023-12-07T16:00:11.105023Z"Novello, Tiago"https://zbmath.org/authors/?q=ai:novello.tiago"da Silva, Vinícius"https://zbmath.org/authors/?q=ai:da-silva.vinicius"Velho, Luiz"https://zbmath.org/authors/?q=ai:velho.luizThe book under review presents a series of algorithms and techniques in computer graphics to visualize Thurston's geometries intrinsically, with the purpose of using them in virtual reality. The ultimate goal is to develop virtual reality in geometric 3-manifolds and orbifolds. Among other ideas, it develops a ray tracing in Riemannian manifolds, by means of geodesics. Several algorithms are described and their performances are compared in several examples, including the isotropic geometries (Euclidean, spherical and hyperbolic) and some non isotropic (Nil, SL(2,R) and Sol).
Reviewer: Joan Porti (Bellaterra)On the cardinality of future worldlines in discrete spacetime structureshttps://zbmath.org/1522.810122023-12-07T16:00:11.105023Z"Çevik, Ahmet"https://zbmath.org/authors/?q=ai:cevik.ahmet-sinan"Seskir, Zeki"https://zbmath.org/authors/?q=ai:seskir.zeki-cSummary: We give an analysis over a variation of causal sets where the light cone of an event is represented by finitely branching trees with respect to any given arbitrary dynamics. We argue through basic topological properties of Cantor space that under certain assumptions about the universe, spacetime structure and causation, given any event \(x\), the number of all possible future worldlines of \(x\) within the many-worlds interpretation is uncountable. However, if all worldlines extending the event \(x\) are `eventually deterministic', then the cardinality of the set of future worldlines with respect to \(x\) is exactly \(\aleph_0\), i.e., countably infinite. We also observe that if there are countably many future worldlines with respect to \(x\), then at least one of them must be necessarily `decidable' in the sense that there is an algorithm which determines whether or not any given event belongs to that worldline. We then show that if there are only finitely many worldlines in the future of an event \(x\), then they are all decidable. We finally point out the fact that there can be only countably many terminating worldlines.Qubit geodesics on the Bloch sphere from optimal-speed Hamiltonian evolutionshttps://zbmath.org/1522.810542023-12-07T16:00:11.105023Z"Cafaro, Carlo"https://zbmath.org/authors/?q=ai:cafaro.carlo"Alsing, Paul M."https://zbmath.org/authors/?q=ai:alsing.paul-mSummary: In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.Sectional curvatures distribution of complexity geometryhttps://zbmath.org/1522.811352023-12-07T16:00:11.105023Z"Wu, Qi-Feng"https://zbmath.org/authors/?q=ai:wu.qifengSummary: In the geometric approach to defining complexity, operator complexity is defined as the distance in the operator space. In this paper, based on the analogy with the circuit complexity, the operator size is adopted as the metric of the operator space where the path length is the complexity. The typical sectional curvatures of this complexity geometry are positive. It is further proved that the typical sectional curvatures are always positive if the metric is an arbitrary function of operator size, while complexity geometry is usually expected to be defined on negatively curved manifolds. By analyzing the sectional curvatures distribution for the \(N\)-qubit system, it is shown that surfaces generated by Hamiltonians of size smaller than the typical size can have negative curvatures. In the large \(N\) limit, the form of complexity metric is uniquely constrained up to constant corrections if we require sectional curvatures are of order \(1/N^2\). With the knowledge of states, the operator size should be modified due to the redundant action of operators, and thus is generalized to be state-dependent. Then we use this state-dependent operator size as the metric of the Hilbert space to define state complexity. It can also be shown that in the Hilbert space, 2-surfaces generated by operators of size much smaller than the typical size acting on typical states also have negative curvatures.Geodesics in the extended Kähler cone of Calabi-Yau threefoldshttps://zbmath.org/1522.812812023-12-07T16:00:11.105023Z"Brodie, Callum R."https://zbmath.org/authors/?q=ai:brodie.callum-r"Constantin, Andrei"https://zbmath.org/authors/?q=ai:constantin.andrei"Lukas, Andre"https://zbmath.org/authors/?q=ai:lukas.andre"Ruehle, Fabian"https://zbmath.org/authors/?q=ai:ruehle.fabianSummary: We present a detailed study of the effective cones of Calabi-Yau threefolds with \(h^{1,1} = 2\), including the possible types of walls bounding the Kähler cone and a classification of the intersection forms arising in the geometrical phases. For all three normal forms in the classification we explicitly solve the geodesic equation and use this to study the evolution near Kähler cone walls and across flop transitions in the context of M-theory compactifications. In the case where the geometric regime ends at a wall beyond which the effective cone continues, the geodesics ``crash'' into the wall, signaling a breakdown of the M-theory supergravity approximation. For illustration, we characterise the structure of the extended Kähler and effective cones of all \(h^{1,1} = 2\) threefolds from the CICY and Kreuzer-Skarke lists, providing a rich set of examples for studying topology change in string theory. These examples show that all three cases of intersection form are realised and suggest that isomorphic flops and infinite flop sequences are common phenomena.Magneto-dimensional resonance on curved surfaceshttps://zbmath.org/1522.812842023-12-07T16:00:11.105023Z"Vybornyi, E. V."https://zbmath.org/authors/?q=ai:vybornyi.e-vThe author is considering a Schrödinger operator of a charge motion in a strong external magnetic field in the resonant case when the cyclotron frequency of the quadratic confinement potential holds the charge near the surface. Using the quantum averaging over the cyclotron and confinement oscillations he is reducing the 3-dimensional magnetic Schrödinger operator to an effective 2-dimensional operator, which determines the small splitting of Landau levels and the quantization of corresponding current on the surface. One obtains an explicit expression for the second order correction induced by the surface curvature to the lowest Landau level near a nondegenerate stationary point of the surface and the corresponding closed curves of the geometric current on the surface.
Reviewer: Alex B. Gaina (Chişinău)Doubled space and extended supersymmetryhttps://zbmath.org/1522.813222023-12-07T16:00:11.105023Z"Blair, Chris D. A."https://zbmath.org/authors/?q=ai:blair.chris-d-a"Hulik, Ondrej"https://zbmath.org/authors/?q=ai:hulik.ondrej"Sevrin, Alexander"https://zbmath.org/authors/?q=ai:sevrin.alexander"Thompson, Daniel C."https://zbmath.org/authors/?q=ai:thompson.daniel-cSummary: The doubled formulation of the worldsheet provides a description of string theory in which T-duality is promoted to a manifest symmetry. Here we extend this approach to \(\mathcal{N} = (2, 2)\) superspace providing a doubled formulation for bi-Hermitian/generalised Kähler target spaces. The theory is described by a single function, a doubled-generalised Kähler potential, supplemented with a manifestly \(\mathcal{N} = (2, 2)\) constraint. Several examples serve to illustrate this construction, including a discussion of the \(\mathcal{N} = (2, 2)\) description of T-folds.Dual D-brane actions in nonrelativistic string theoryhttps://zbmath.org/1522.813462023-12-07T16:00:11.105023Z"Ebert, Stephen"https://zbmath.org/authors/?q=ai:ebert.stephen"Sun, Hao-Yu"https://zbmath.org/authors/?q=ai:sun.hao-yu"Yan, Ziqi"https://zbmath.org/authors/?q=ai:yan.ziqiSummary: We study worldvolume actions for D-branes coupled to the worldvolume U(1) gauge field and Ramond-Ramond (RR) potentials in nonrelativistic string theory. This theory is a self-contained corner of relativistic string theory and has a string spectrum with a Galilean-invariant dispersion relation. We therefore refer to such D-branes in nonrelativistic string theory as nonrelativistic D-branes. We focus on the bosonic fields in spacetime and also couple the D-branes to general closed string geometry, Kalb-Ramond, and dilaton background fields. We dualize nonrelativistic D-branes by performing a duality transformation on the worldvolume U(1) gauge field and uncover novel dual D-brane actions. This generalizes familiar properties, such as the \(\mathrm{SL}(2, \mathbb{Z})\) duality in Type IIB superstring theory and the relation between Type IIA superstring and M-theory, to nonrelativistic string and M-theory. Moreover, we generalize the limit of string theory, in which nonrelativistic string theory arises, to include RR potentials. This stringy limit induces a codimension-two foliation structure in spacetime. This spacetime geometry is non-Riemannian and known as string Newton-Cartan geometry. In contrast, nonrelativistic M-theory that we probe by dualizing D2- and D4-branes in nonrelativistic string theory arises as a membrane limit of M-theory, and it is coupled to a membrane Newton-Cartan geometry with a codimension-three foliation structure. We also discuss T-duality in nonrelativistic string theory and generalize Buscher rules from earlier work to include RR potentials.2d \(\mathcal{N} = (0, 1)\) gauge theories and Spin(7) orientifoldshttps://zbmath.org/1522.813542023-12-07T16:00:11.105023Z"Franco, Sebastián"https://zbmath.org/authors/?q=ai:franco.sebastian"Mininno, Alessandro"https://zbmath.org/authors/?q=ai:mininno.alessandro"Uranga, Ángel M."https://zbmath.org/authors/?q=ai:uranga.angel-m"Yu, Xingyang"https://zbmath.org/authors/?q=ai:yu.xingyangSummary: We initiate the geometric engineering of 2d \(\mathcal{N} = (0, 1)\) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as \textit{Spin(7) orientifolds}. Spin(7) orientifolds explicitly realize the perspective on 2d \(\mathcal{N} = (0, 1)\) theories as real slices of \(\mathcal{N} = (0, 2)\) ones. Remarkably, this projection is geometrically realized as Joyce's construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions.Heterotic Kerr-schild double field theory and its double Yang-Mills formulationhttps://zbmath.org/1522.813832023-12-07T16:00:11.105023Z"Lescano, Eric"https://zbmath.org/authors/?q=ai:lescano.eric"Roychowdhury, Sourav"https://zbmath.org/authors/?q=ai:roychowdhury.souravSummary: We present a formulation of heterotic Double Field Theory (DFT), where the fundamental fields are in \(O(D, D)\) representations. The theory is obtained splitting an \(O(D, D + K)\) duality invariant DFT. This procedure produces a Green-Schwarz mechanism for the generalized metric, and a fundamental gauge field which transforms as a gauge connection only to leading order. After parametrization, the former induces a non-covariant transformation on the metric tensor, which can be removed considering field redefinitions, and an ordinary Green-Schwarz mechanism on the b-field. Within this framework we explore perturbative properties of heterotic DFT. We use a relaxed version of the generalized Kerr-Schild ansatz (GKSA), where the generalized background metric is perturbed up to quadratic order considering a single null vector and the gauge field is linearly perturbed before parametrization. Finally we compare the dynamics of the gauge field and the generalized metric in order to inspect the behavior of the classical double copy correspondence at the DFT level.Beyond second-moment approximation in fuzzy-field-theory-like matrix modelshttps://zbmath.org/1522.814162023-12-07T16:00:11.105023Z"Šubjaková, Mária"https://zbmath.org/authors/?q=ai:subjakova.maria"Tekel, Juraj"https://zbmath.org/authors/?q=ai:tekel.jurajSummary: We investigate the phase structure of a special class of multi-trace hermitian matrix models, which are candidates for the description of scalar field theory on fuzzy spaces. We include up to the fourth moment of the eigenvalue distribution into the multi-trace part of the probability distribution, which stems from the kinetic term of the field theory action. We show that by considering different multi-trace behavior in the large moment and in the small moment regimes of the model, it is possible to obtain a matrix model, which describes the numerically observed phase structure of fuzzy field theories. Including the existence of uniform order phase, triple point, and an approximately straight transition line between the uniform and non-uniform order phases.Galilean gauge theories from null reductionshttps://zbmath.org/1522.814462023-12-07T16:00:11.105023Z"Bagchi, Arjun"https://zbmath.org/authors/?q=ai:bagchi.arjun"Basu, Rudranil"https://zbmath.org/authors/?q=ai:basu.rudranil"Islam, Minhajul"https://zbmath.org/authors/?q=ai:islam.minhajul"Kolekar, Kedar S."https://zbmath.org/authors/?q=ai:kolekar.kedar-s"Mehra, Aditya"https://zbmath.org/authors/?q=ai:mehra.adityaSummary: The procedure of null reduction provides a concrete way of constructing field theories with Galilean invariance. We use this to examine Galilean gauge theories, viz. Galilean electrodynamics and Yang-Mills theories in spacetime dimensions 3 and 4. Different non-relativistic conformal symmetries arise in these contexts: Schrödinger symmetry in \(d = 3\) and Galilean conformal symmetry in \(d = 4\). A canonical analysis further reveals that the symmetries enhance to their infinite dimensional versions in phase space and pick up central extensions. In addition, for the Abelian theory, we discuss non-relativistic electro- magnetic duality in \(d = 3\) and its difference with the \(d = 4\) version. We also mention some quantum aspects for both Abelian and non-Abelian theories.Information geometry and holographic correlatorshttps://zbmath.org/1522.814582023-12-07T16:00:11.105023Z"Bohra, Hardik"https://zbmath.org/authors/?q=ai:bohra.hardik"Kakkar, Ashish"https://zbmath.org/authors/?q=ai:kakkar.ashish"Sivaramakrishnan, Allic"https://zbmath.org/authors/?q=ai:sivaramakrishnan.allicSummary: We explore perturbative corrections to quantum information geometry. In particular, we study a Bures information metric naturally associated with the correlation functions of a conformal field theory. We compute the metric of holographic four-point functions and include corrections generated by tree Witten diagrams in the bulk. In this setting, we translate properties of correlators into the language of information geometry. Cross terms in the information metric encode non-identity operators in the OPE. We find that the information metric is asymptotically AdS. Finally, we discuss an information metric for transition amplitudes.Consistent and non-consistent deformations of gravitational theorieshttps://zbmath.org/1522.815482023-12-07T16:00:11.105023Z"Barbero G., J. Fernando"https://zbmath.org/authors/?q=ai:barbero-g.j-fernando"Basquens, Marc"https://zbmath.org/authors/?q=ai:basquens.marc"Díaz, Bogar"https://zbmath.org/authors/?q=ai:diaz.bogar"Villaseñor, Eduardo J. S."https://zbmath.org/authors/?q=ai:villasenor.eduardo-j-sSummary: We study the internally abelianized version of a range of gravitational theories, written in connection tetrad form, and study the possible interaction terms that can be added to them in a consistent way. We do this for \(2+1\) and \(3+1\) dimensional models. In the latter case we show that the Cartan-Palatini and Holst actions are not consistent deformations of their abelianized versions. We also show that the Husain-Kuchař and Euclidean self-dual actions are consistent deformations of their abelianized counterparts. This suggests that if the latter can be quantized, it could be possible to devise a perturbative scheme leading to the quantization of Euclidean general relativity along the lines put forward by \textit{L. Smolin} [Classical Quantum Gravity 9, No. 4, 883--893 (1992; Zbl 0747.58064)] in the early nineties.Index-like theorem for massless fermions in spherically symmetric monopole backgroundshttps://zbmath.org/1522.815502023-12-07T16:00:11.105023Z"Brennan, T. Daniel"https://zbmath.org/authors/?q=ai:brennan.t-danielSummary: In this paper we study massless fermions coupled to spherically symmetric \(\mathrm{SU}(N)\) monopoles without Yukawa couplings between the Higgs and fermion fields. The corresponding Dirac operator is not Fredholm and the associated eigenfunctions are not \(L^2\)-normalizable. Here we derive a formula for the dimension of the plane-wave normalizable kernel of such a Dirac operator for fermions of any representation of \(\mathrm{SU}(N)\) in the presence of any spherically symmetric monopole background. Notably, our results also apply to fermions coupled to monopoles that preserve non-abelian gauge symmetry.From \(\beta\) to \(\eta \): a new cohomology for deformed Sasaki-Einstein manifoldshttps://zbmath.org/1522.816722023-12-07T16:00:11.105023Z"Tasker, Edward Lødøen"https://zbmath.org/authors/?q=ai:tasker.edward-lodoenSummary: We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups \(H_{\bar{\partial}}^{(p, 0)}(k)\) graded by their charge under the Reeb vector. We then introduce a new cohomology, \(\eta\)-cohomology, which is defined by a CR structure and a holomorphic function \(f\) with non-vanishing \(\eta\equiv\mathrm{d}f\). It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the \(4d\) \(\mathcal{N} = 1\) SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the \(\eta\)-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the \(\eta\)-cohomology groups. We verify that this relation is satisfied in the case of \(\mathrm{S}^5\), and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of \(\mathbb{P}^2\), \(\mathbb{P}^1\times\mathbb{P}^1\) and the del Pezzo surfaces.Neural network approximations for Calabi-Yau metricshttps://zbmath.org/1522.830012023-12-07T16:00:11.105023Z"Jejjala, Vishnu"https://zbmath.org/authors/?q=ai:jejjala.vishnu"Mayorga Peña, Damián Kaloni"https://zbmath.org/authors/?q=ai:mayorga-pena.damian-kaloni"Mishra, Challenger"https://zbmath.org/authors/?q=ai:mishra.challengerSummary: Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for K3, the Fermat quintic, and the Dwork quintic. This investigation employs a simple, modular neural network architecture that is capable of approximating Ricci flat Kähler metrics for Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness and consistency of the metric decrease after training. This improvement is corroborated by the performance of the trained network on an independent validation set. Finally, we demonstrate the consistency of the learnt metric by showing that it is invariant under the discrete symmetries it is expected to possess.Charges and fluxes on (perturbed) non-expanding horizonshttps://zbmath.org/1522.830052023-12-07T16:00:11.105023Z"Ashtekar, Abhay"https://zbmath.org/authors/?q=ai:ashtekar.abhay"Khera, Neev"https://zbmath.org/authors/?q=ai:khera.neev"Kolanowski, Maciej"https://zbmath.org/authors/?q=ai:kolanowski.maciej"Lewandowski, Jerzy"https://zbmath.org/authors/?q=ai:lewandowski.jerzySummary: In a companion paper [\textit{A. Ashtekar} et al., J. High Energy Phys. 2022, No. 1, Paper No. 28, 33 p. (2022; Zbl 1521.83082)] we showed that the symmetry group \(\mathfrak{G}\) of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group \(\mathfrak{B}\) at \(\mathcal{I}^+\). For each infinitesimal generator of \(\mathfrak{G}\), we now define a charge and a flux on NEHs \textit{as well as perturbed} NEHs. The procedure uses the covariant phase space framework in presence of internal null boundaries \(\mathcal{N}\) along the lines of [\textit{V. Chandrasekaran} et al., J. High Energy Phys. 2018, No. 11, Paper No. 125, 68 p. (2018; Zbl 1404.83016); \textit{V. Chandrasekaran} and \textit{K. Prabhu}, J. High Energy Phys. 2019, No. 10, Paper No. 229, 27 p. (2019; Zbl 1427.83007); \textit{R. Oliveri} and \textit{S. Speziale}, Gen. Relativ. Gravitation 52, No. 8, Paper No. 83, 51 p. (2020; Zbl 1468.83009); \textit{W. Wieland}, J. High Energy Phys. 2021, No. 4, Paper No. 95, 53 p. (2021; Zbl 1462.83010); \textit{L. Freidel} et al., J. High Energy Phys. 2021, No. 9, Paper No. 83, 38 p. (2021; Zbl 1472.83012)]. However, \(\mathcal{N}\) is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of \(\mathfrak{G}\) are free of physically unsatisfactory features that can arise if \(\mathcal{N}\) is allowed to be a general null boundary. In particular, all fluxes vanish if \(\mathcal{N}\) is an NEH, just as one would hope; and fluxes associated with symmetries representing `time-translations' are positive definite on perturbed NEHs. These results hold for zero as well as non-zero cosmological constant. In the asymptotically flat case, as noted in [Ashtekar et al., loc. cit.], \(\mathcal{I}^\pm\) \textit{are NEHs in the conformally completed space-time} but with an extra structure that reduces \(\mathfrak{G}\) to \(\mathfrak{B}\). The flux expressions at \(\mathcal{N}\) reflect this synergy between NEHs and \(\mathcal{I}^+\). In a forthcoming paper, this close relation between NEHs and \(\mathcal{I}^+\) will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at \(\mathcal{I}^+\).Einstein's equations on a 4-manifold of conformal torsion-free connectionhttps://zbmath.org/1522.830112023-12-07T16:00:11.105023Z"Krivonosov, Leonid N."https://zbmath.org/authors/?q=ai:krivonosov.leonid-nikolaevich"Luk'yanov, Vyacheslav A."https://zbmath.org/authors/?q=ai:lukyanov.vyacheslav-anatolevichSummary: The main defect of Einstein equations -- non geometrical right part -- is eliminated. The key concept of equidual tensor is introduced. It appeared to be in a close relation both with Einstein's equations, and with Yang-Mills equations. The criterion of equidual basic affinor of conformal connection manifold without torsion is received. Decomposition of the basic affinor into a sum of equidual, conformally invariant and irreducible summands is found. \textit{A. Z. Petrov}'s algebraic classification [New methods in general relativity theory (Russian). Moscow: Nauka (1966; Zbl 0146.23901)] is generalized. Einstein equations are given a new variational foundation and their geometrical nature is found. Geometric sense of acceleration and dilatation gauge transformations is specified.Ambiguity resolution for integrable gravitational chargeshttps://zbmath.org/1522.830122023-12-07T16:00:11.105023Z"Speranza, Antony J."https://zbmath.org/authors/?q=ai:speranza.antony-jSummary: Recently, \textit{L. Ciambelli}, \textit{R. G. Leigh}, and \textit{P.-C. Pai} (CLP) [``Embeddings and integrable charges for extended corner symmetry'', Preprint, \url{arXiv:2111.13181}] have shown that nonzero charges integrating Hamilton's equation can be defined for all diffeomorphisms acting near the boundary of a subregion in a gravitational theory. This is done by extending the phase space to include a set of embedding fields that parameterize the location of the boundary. Because their construction differs from previous works on extended phase spaces by a covariant phase space ambiguity, the question arises as to whether the resulting charges are unambiguously defined. Here, we demonstrate that ambiguity-free charges can be obtained by appealing to the variational principle for the subregion, following recent developments on dealing with boundaries in the covariant phase space. Resolving the ambiguity produces corrections to the diffeomorphism charges, and also generates additional obstructions to integrability of Hamilton's equation. We emphasize the fact that the CLP extended phase space produces nonzero diffeomorphism charges distinguishes it from previous constructions in which diffeomorphisms are pure gauge, since the embedding fields can always be eliminated from the latter by a choice of unitary gauge. Finally, we show that Wald-Zoupas charges, with their characteristic obstruction to integrability, are associated with a modified transformation in the extended phase space, clarifying the reason behind integrability of Hamilton's equation for standard diffeomorphisms.Limits of spacetimeshttps://zbmath.org/1522.830262023-12-07T16:00:11.105023Z"Geroch, Robert"https://zbmath.org/authors/?q=ai:geroch.robertSummary: The limits of a one-parameter family of spacetimes are defined, and the properties of such limits discussed. The definition is applied to an investigation of the Schwarzschild solution as a limit of the Reissner-Nordström solution as the charge parameter goes to zero. Two new techniques -- rigidity of a geometrical structure and Killing transport -- are introduced. Several applications of these two subjects, both to limits and to certain other questions in differential geometry, are discussed.Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativityhttps://zbmath.org/1522.830292023-12-07T16:00:11.105023Z"Prabhu, Kartik"https://zbmath.org/authors/?q=ai:prabhu.kartik"Shehzad, Ibrahim"https://zbmath.org/authors/?q=ai:shehzad.ibrahimSummary: We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that \textit{all} Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacetimes, proving a relationship that was conjectured by Strominger. Assuming additional suitable conditions are satisfied at timelike infinities, this proves that the flux of all BMS charges is conserved in any classical gravitational scattering process in these spacetimes.Bound on quantum fluctuations in gravitational waves from LIGO-virgohttps://zbmath.org/1522.830342023-12-07T16:00:11.105023Z"Hertzberg, Mark P."https://zbmath.org/authors/?q=ai:hertzberg.mark-p"Litterer, Jacob A."https://zbmath.org/authors/?q=ai:litterer.jacob-a(no abstract)Stability of non-degenerate Ricci-type Palatini theorieshttps://zbmath.org/1522.830392023-12-07T16:00:11.105023Z"Annala, Jaakko"https://zbmath.org/authors/?q=ai:annala.jaakko"Räsänen, Syksy"https://zbmath.org/authors/?q=ai:rasanen.syksy(no abstract)Positivity of mass in higher dimensionshttps://zbmath.org/1522.830402023-12-07T16:00:11.105023Z"Cameron, Peter"https://zbmath.org/authors/?q=ai:cameron.peter-j|cameron.peterSummary: The positive mass theorem in higher dimensions is proved using causality arguments inspired by those of Penrose et al. (a positive mass theorem based on the focusing and retardation of null geodesics, 1993) in \(3+1\) dimensions.Local continuity of angular momentum and Noether charge for matter in general relativityhttps://zbmath.org/1522.830412023-12-07T16:00:11.105023Z"Croft, Robin"https://zbmath.org/authors/?q=ai:croft.robinSummary: Conservation laws have many applications in numerical relativity. However, it is not straightforward to define local conservation laws for general dynamic spacetimes due the lack of coordinate translation symmetries. In flat space, the rate of change of energy-momentum within a finite spacelike volume is equivalent to the flux integrated over the surface of this volume; for general spacetimes it is necessary to include a volume integral of a source term arising from spacetime curvature. In this work a study of continuity of matter in general relativity is extended to include angular momentum of matter and Noether currents associated with gauge symmetries. Expressions for the Noether charge and flux of complex scalar fields and complex Proca fields are found using this formalism. Expressions for the angular momentum density, flux and source are also derived which are then applied to a numerical relativity collision of boson stars in 3D with non-zero impact parameter as an illustration of the methods.Imprint of galactic rotation curves and metric fluctuations on the recombination era anisotropyhttps://zbmath.org/1522.830452023-12-07T16:00:11.105023Z"Mannheim, Philip D."https://zbmath.org/authors/?q=ai:mannheim.philip-dSummary: In applications of the conformal gravity theory it has been shown that a scale of order 105 Mpc due to large scale inhomogeneities such as clusters of galaxies is imprinted on the rotation curves of galaxies. Here we show that this same scale is imprinted on recombination era anisotropies in the cosmic microwave background. We revisit an analysis due to Mannheim and Horne, to show that in the conformal gravity theory the particle horizon distance scale of metric signals that originate in the primordial nucleosynthesis era at \(10^9\)\,\(^\circ\mathrm{K}\) can encompass the entire recombination era sky. Similarly, the particle horizon distance scale of acoustic signals that originate at \(10^{13}\)\,\(^\circ\mathrm{K}\) can also encompass the entire recombination era sky. We show that the amplitudes of metric fluctuations that originate in the nucleosynthesis era can grow by a factor of \(10^{12}\) by recombination, and by a factor of \(10^{18}\) by the current time. In addition we find that without any period of exponential expansion a fluctuation amplitude that begins at a temperature of order \(10^{33}\)\,\(^\circ\mathrm{K}\) can grow by a factor of \(10^{60}\) by recombination and by a factor of \(10^{66}\) by the current time.Particle-like solutions in the generalized SU(2) Proca theoryhttps://zbmath.org/1522.830462023-12-07T16:00:11.105023Z"Martínez, Jhan N."https://zbmath.org/authors/?q=ai:martinez.jhan-n"Rodríguez, José F."https://zbmath.org/authors/?q=ai:rodriguez.jose-felix"Rodríguez, Yeinzon"https://zbmath.org/authors/?q=ai:rodriguez.yeinzon"Gómez, Gabriel"https://zbmath.org/authors/?q=ai:gomez.gabriel(no abstract)Teleparallel Newton-Cartan gravityhttps://zbmath.org/1522.830502023-12-07T16:00:11.105023Z"Schwartz, Philip K."https://zbmath.org/authors/?q=ai:schwartz.philip-kSummary: We discuss a teleparallel version of Newton-Cartan gravity. This theory arises as a formal large-speed-of-light limit of the teleparallel equivalent of general relativity (TEGR). Thus, it provides a geometric formulation of the Newtonian limit of TEGR, similar to standard Newton-Cartan gravity being the Newtonian limit of general relativity. We show how by a certain gauge-fixing the standard formulation of Newtonian gravity can be recovered.Double scaling limits of Dirac ensembles and Liouville quantum gravityhttps://zbmath.org/1522.830702023-12-07T16:00:11.105023Z"Hessam, Hamed"https://zbmath.org/authors/?q=ai:hessam.hamed"Khalkhali, Masoud"https://zbmath.org/authors/?q=ai:khalkhali.masoud"Pagliaroli, Nathan"https://zbmath.org/authors/?q=ai:pagliaroli.nathanSummary: In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.The constraints of post-quantum classical gravityhttps://zbmath.org/1522.830762023-12-07T16:00:11.105023Z"Oppenheim, Jonathan"https://zbmath.org/authors/?q=ai:oppenheim.jonathan"Weller-Davies, Zachary"https://zbmath.org/authors/?q=ai:weller-davies.zacharySummary: We study a class of theories in which space-time is treated classically, while interacting with quantum fields. These circumvent various no-go theorems and the pathologies of semi-classical gravity, by being linear in the density matrix and phase-space density. The theory can either be considered fundamental or as an effective theory where the classical limit is taken of space-time. The theories have the dynamics of general relativity as their classical limit and provide a way to study the back-action of quantum fields on the space-time metric. The theory is invariant under spatial diffeomorphisms, and here, we provide a methodology to derive the constraint equations of such a theory by imposing invariance of the dynamics under time-reparametrization invariance. This leads to generalisations of the Hamiltonian and momentum constraints. We compute the constraint algebra for a wide class of realisations of the theory (the ``discrete class'') in the case of a quantum scalar field interacting with gravity. We find that the algebra doesn't close without additional constraints, although these do not necessarily reduce the number of local degrees of freedom.Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravityhttps://zbmath.org/1522.830802023-12-07T16:00:11.105023Z"Trugenberger, C. A."https://zbmath.org/authors/?q=ai:trugenberger.carlo-aSummary: Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert action formulated on an ensemble of random graphs. There is strong evidence for a second-order quantum phase transition separating a random phase at strong coupling from an ordered, geometric phase at weak coupling. Here we derive the picture of space-time that emerges in the geometric phase, given such a continuous phase transition. In the geometric phase, ground-state graphs are discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On such manifolds, diffusion is ballistic. Asymptotically, diffusion time is soldered with a manifold coordinate and, consequently, the probability distribution is governed by the wave equation on the corresponding Lorentzian manifold of positive curvature, de Sitter space-time. With this asymptotic Lorentzian picture, the original negative curvature of the Riemannian manifold turns into a positive cosmological constant. The Lorentzian picture, however, is valid only asymptotically and cannot be extrapolated back in coordinate time. Before a certain epoch, coordinate time looses its meaning and the universe is a negative-curvature Riemannian ``shuttlecock'' with ballistic diffusion, thereby avoiding a big bang singularity. The emerging coordinate time leads to a de Sitter version of the holographic principle relating the bulk isometries with boundary conformal transformations. While the topological boundary dimension is \((D-1)\), the so-called ``dimension at infinity'' of negative curvature manifolds, i.e. the large-scale spectral dimension seen by diffusion processes with no spectral gap, those that can probe the geometry at infinity, is always three.Constraining Weil-Petersson volumes by universal random matrix correlations in low-dimensional quantum gravityhttps://zbmath.org/1522.830822023-12-07T16:00:11.105023Z"Weber, Torsten"https://zbmath.org/authors/?q=ai:weber.torsten"Haneder, Fabian"https://zbmath.org/authors/?q=ai:haneder.fabian"Richter, Klaus"https://zbmath.org/authors/?q=ai:richter.klaus-jurgen"Urbina, Juan Diego"https://zbmath.org/authors/?q=ai:urbina.juan-diegoSummary: Based on the discovery of the duality between Jackiw-Teitelboim quantum gravity and a double-scaled matrix ensemble by \textit{P. Saad} et al. in [``JT gravity as a matrix integral'', Preprint, \url{arXiv:1903.11115}], we show how consistency between the two theories in the universal random matrix theory (RMT) limit imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil-Petersson (WP) volumes, solving a celebrated nonlinear recursion formula that is notoriously difficult to analyse. Since our results imply \textit{linear} relations between the coefficients of the WP volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction. In this way, we propose a long-term program to improve the understanding of mathematically hard aspects concerning moduli spaces of hyperbolic manifolds by using universal RMT results as input.Curvature blow-up and mass inflation in spherically symmetric collapse to a Schwarzschild black holehttps://zbmath.org/1522.831112023-12-07T16:00:11.105023Z"An, Xinliang"https://zbmath.org/authors/?q=ai:an.xinliang"Gajic, Dejan"https://zbmath.org/authors/?q=ai:gajic.dejanSummary: We study the black hole interiors of spacetimes arising from gravitational collapse within the spherically symmetric Einstein-scalar field system. By investigating the precise blow-up rates of curvature and mass at the spacelike singularity, near timelike infinity, we give an answer to whether the interior metric converges to a Schwarzschild metric. We show, in particular, that the Kretschmann scalar blows up faster than in the Schwarzschild setting, due to mass inflation. Moreover, the blow-up rate is not constant and converges to the Schwarzschild rate towards timelike infinity and it depends on the precise late-time polynomial behaviour of the scalar field along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism.On acceleration in three dimensionshttps://zbmath.org/1522.831152023-12-07T16:00:11.105023Z"Arenas-Henriquez, Gabriel"https://zbmath.org/authors/?q=ai:arenas-henriquez.gabriel"Gregory, Ruth"https://zbmath.org/authors/?q=ai:gregory.ruth"Scoins, Andrew"https://zbmath.org/authors/?q=ai:scoins.andrewSummary: We go ``back to basics'', studying accelerating systems in \(2 + 1\) AdS gravity \textit{ab initio}. We find three classes of geometry, which we interpret by studying holographically their physical parameters. From these, we construct stationary, accelerating point particles; one-parameter extensions of the BTZ family resembling an accelerating black hole; and find new solutions including a novel accelerating ``BTZ geometry'' not continuously connected to the BTZ black hole as well as some black funnel solutions.Exotic marginally outer trapped surfaces in rotating spacetimes of any dimensionhttps://zbmath.org/1522.831342023-12-07T16:00:11.105023Z"Booth, Ivan"https://zbmath.org/authors/?q=ai:booth.ivan-s"Chan, Kam To Billy"https://zbmath.org/authors/?q=ai:chan.kam-to-billy"Hennigar, Robie A."https://zbmath.org/authors/?q=ai:hennigar.robie-a"Kunduri, Hari"https://zbmath.org/authors/?q=ai:kunduri.hari-k"Muth, Sarah"https://zbmath.org/authors/?q=ai:muth.sarahSummary: The recently developed MOTSodesic method for locating marginally outer trapped surfaces (MOTSs) was effectively restricted to non-rotating spacetimes. In this paper we extend the method to include (multi-)axisymmetric time slices of (multi-)axisymmetric spacetimes of any dimension. We then apply this method to study MOTSs in the BTZ, Kerr and Myers-Perry black holes. While there are many similarities between the MOTSs observed in these spacetimes and those seen in Schwarzschild and Reissner-Nordström, details of the more complicated geometries also introduce some new, previously unseen, behaviours.Toroidal tidal effects in microstate geometrieshttps://zbmath.org/1522.831392023-12-07T16:00:11.105023Z"Čeplak, Nejc"https://zbmath.org/authors/?q=ai:ceplak.nejc"Hampton, Shaun"https://zbmath.org/authors/?q=ai:hampton.shaun-d"Li, Yixuan"https://zbmath.org/authors/?q=ai:li.yixuanSummary: Tidal effects in capped geometries computed in previous literature display no dynamics along internal (toroidal) directions. However, the dual CFT picture suggests otherwise. To resolve this tension, we consider a set of infalling null geodesics in a family of black hole microstate geometries with a smooth cap at the bottom of a long BTZ-like throat. Using the Penrose limit, we show that a string following one of these geodesics feels tidal stresses along all spatial directions, including internal toroidal directions. We find that the tidal effects along the internal directions are of the same order of magnitude as those along other, non-internal, directions. Furthermore, these tidal effects oscillate as a function of the distance from the cap -- as a string falls down the throat it alternately experiences compression and stretching. We explain some physical properties of this oscillation and comment on the dual CFT interpretation.Gravitational collapse and formation of a black hole in a type II minimally modified gravity theoryhttps://zbmath.org/1522.831582023-12-07T16:00:11.105023Z"De Felice, Antonio"https://zbmath.org/authors/?q=ai:de-felice.antonio"Maeda, Kei-ichi"https://zbmath.org/authors/?q=ai:maeda.keiichi"Mukohyama, Shinji"https://zbmath.org/authors/?q=ai:mukohyama.shinji"Pookkillath, Masroor C."https://zbmath.org/authors/?q=ai:pookkillath.masroor-c(no abstract)Einstein's hole argument and Schwarzschild singularitieshttps://zbmath.org/1522.831732023-12-07T16:00:11.105023Z"Gogberashvili, Merab"https://zbmath.org/authors/?q=ai:gogberashvili.merabSummary: According to the Einstein hole argument, vacuum metric solutions are equivalent only if they correspond to the same energy-momentum tensor in the source region. In this paper it is shown that singular coordinates that are used to show Schwarzschild geodesics completeness, introduce the fictive delta-like sources at the horizon. Then, metric tensors obtained by such singular transformations, cannot be considered as solutions of the same Einstein equations with the central source.The rules of 4-dimensional perspective: how to implement Lorentz transformations in relativistic visualizationhttps://zbmath.org/1522.831832023-12-07T16:00:11.105023Z"Hamilton, Andrew J. S."https://zbmath.org/authors/?q=ai:hamilton.andrew-j-sSummary: This paper presents a pedagogical introduction to the issue of how to implement Lorentz transformations in relativistic visualization. The most efficient approach is to use the even geometric algebra in \(3+1\) spacetime dimensions, or equivalently complex quaternions, which are fast, compact, and robust, and straightforward to compose, interpolate, and spline. The approach has been incorporated into the Black Hole Flight Simulator, an interactive general relativistic ray-tracing program developed by the author.Thermal analysis with emission energy of perturbed black hole in \(f(Q)\) gravityhttps://zbmath.org/1522.831972023-12-07T16:00:11.105023Z"Javed, Faisal"https://zbmath.org/authors/?q=ai:javed.faisal"Mustafa, G."https://zbmath.org/authors/?q=ai:mustafa.ghulam"Mumtaz, Saadia"https://zbmath.org/authors/?q=ai:mumtaz.saadia"Atamurotov, Farruh"https://zbmath.org/authors/?q=ai:atamurotov.farruhSummary: In the present paper, we investigate the thermodynamics of a new perturbed black hole solution in symmetric teleparallel gravity. Our main focus is on the solution for a nonlinear model like \(Q + \alpha Q^2\). We study new features of null geodesics and quasi-normal modes that occur for the current perturbed black hole solution. We inspect the critical behavior of thermal fluctuation and phase transitions in the background of a considered black hole solution and get the Davies points. To complete this study, we also explore the interesting aspects of the emission energy and quantum fluctuations. We highlight that our results are in good agreement with the thermodynamics of the previous black hole solutions and assumptions presented in the literature.Photon rings around warped black holeshttps://zbmath.org/1522.831992023-12-07T16:00:11.105023Z"Kapec, Daniel"https://zbmath.org/authors/?q=ai:kapec.daniel"Lupsasca, Alexandru"https://zbmath.org/authors/?q=ai:lupsasca.alexandru"Strominger, Andrew"https://zbmath.org/authors/?q=ai:strominger.andrewSummary: The black hole photon ring is a prime target for upcoming space-based VLBI missions seeking to image the fine structure of astrophysical black holes. The classical Lyapunov exponents of the corresponding nearly bound null geodesics control the quasinormal ringing of a perturbed black hole as it settles back down to equilibrium, and they admit a holographic interpretation in terms of quantum Ruelle resonances of the microstate dual to the Kerr black hole. Recent work has identified a number of emergent symmetries related to the intricate self-similar structure of the photon ring. Here, we explore this web of interrelated phenomena in an exactly soluble example that arises as an approximation to the near-extremal Kerr black hole. The self-dual warped \(\mathrm{AdS}_3\) geometry has a photon ring as well as \(\mathsf{SL}(2, \mathbb{R})\) isometries and an exactly calculable quasinormal mode (QNM) spectrum. We show explicitly that the geometric optics approximation reproduces the eikonal limit of the exact scalar QNM spectrum, as well as the approximate `near-ring' wavefunctions. The \(\mathsf{SL}(2, \mathbb{R})\) isometries are directly related to the emergent conformal symmetry of the photon ring in black hole images but are distinct from a recently discussed conformal symmetry of the eikonal QNM spectrum. The equivalence of the classical QNM spectrum -- and thus the photon ring -- to the quantum Ruelle resonances in the context of a spacetime with a putative holographic dual suggests that the photon ring of a warped black hole is indeed part of the black hole hologram.Hawking radiation as tunneling with pressure and volume of the RN-AdS black holehttps://zbmath.org/1522.832322023-12-07T16:00:11.105023Z"Ren, Zhi-Xuan"https://zbmath.org/authors/?q=ai:ren.zhi-xuan"Zeng, Xiao-Xiong"https://zbmath.org/authors/?q=ai:zeng.xiaoxiong"Han, Yi-Wen"https://zbmath.org/authors/?q=ai:han.yiwen"Hu, Cheng"https://zbmath.org/authors/?q=ai:hu.chengSummary: In consideration of the thermodynamic pressure and volume, we present a short and direct derivation of Hawking radiation as a tunneling process for the charged particles. Using Parikh's Semi-classical tunneling method and Lagrangian analysis on the action, we provide the geodesic equation of the massive particles via tunneling from the Anit-de Sitter (AdS) black hole. Special attention is given to calculating the imaginary part before and after particles via the horizon as the pressure and volume are considered. The result shows that the emission rates are always related to the change of Bekenstein-Hawking entropy and the exact spectrum is not precisely thermal, which are consistent with the case without pressure and volume.Dain's invariant for black hole initial datahttps://zbmath.org/1522.832352023-12-07T16:00:11.105023Z"Sansom, R."https://zbmath.org/authors/?q=ai:sansom.r"Valiente Kroon, J. A."https://zbmath.org/authors/?q=ai:valiente-kroon.juan-antonioSummary: Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that approximate symmetries coincide with actual symmetries when they are present allows one to construct geometric invariants characterising the symmetry. In this paper, we extend Dain's construction of geometric invariants characterising stationarity to the case of initial data sets for the Einstein equations corresponding to black hole spacetimes. We prove the existence and uniqueness of solutions to a boundary value problem showing that one can always find approximate Killing vectors in black hole spacetimes and these coincide with actual Killing vectors when they are present. In the time-symmetric setting we make use of a \(2 + 1\) decomposition to construct a geometric invariant on a marginally outer trapped surface that vanishes if and only if the Killing initial data equations are locally satisfied.From spin foams to area metric dynamics to gravitonshttps://zbmath.org/1522.832492023-12-07T16:00:11.105023Z"Dittrich, Bianca"https://zbmath.org/authors/?q=ai:dittrich.bianca"Kogios, Athanasios"https://zbmath.org/authors/?q=ai:kogios.athanasiosSummary: Although spin foams arose as quantizations of the length metric degrees of freedom, the quantum configuration space is rather based on areas as more fundamental variables. This is also highlighted by the semi-classical limit of four-dimensional spin foam models, which is described by the Area Regge action. Despite its central importance to spin foams the dynamics encoded by the Area Regge action is only poorly understood, in particular in the continuum limit. We perform here a systematic investigation of the dynamics defined by the Area Regge action on a regular centrally subdivided hypercubical lattice. This choice of lattice avoids many problems of the non-subdivided hypercubical lattice, for which the Area Regge action is singular. The regularity of the lattice allows to extract the continuum limit and its corrections, order by order in the lattice constant. We show that, contrary to widespread expectations which arose from the so-called flatness problem of spin foams, the continuum limit of the Area Regge action does describe to leading order the same graviton dynamics as general relativity. The next-to-leading order correction to the effective action for the length metric is of second order in the lattice constant, and is given by a quadratic term in the Weyl curvature tensor. This correction can be understood to originate from an underlying dynamics of area metrics. This suggests that the continuum limit of spin foam dynamics does lead to massless gravitons, and that the leading order quantum corrections can be understood to emerge from a generalization of the configuration space from length to area metrics.Stable big bang formation for Einstein's equations: the complete sub-critical regimehttps://zbmath.org/1522.832572023-12-07T16:00:11.105023Z"Fournodavlos, Grigorios"https://zbmath.org/authors/?q=ai:fournodavlos.grigorios"Rodnianski, Igor"https://zbmath.org/authors/?q=ai:rodnianski.igor"Speck, Jared"https://zbmath.org/authors/?q=ai:speck.jaredSummary: For \((t,x) \in (0,\infty )\times \mathbb{T}^{\mathfrak{D}} \), the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as \(t \downarrow 0\), i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents \(\widetilde{q}_1,\cdots ,\widetilde{q}_{\mathfrak{D}} \in \mathbb{R} \), which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at \(\lbrace t = 1 \rbrace \), as long as the exponents are ``sub-critical'' in the following sense: \( \underset{\substack{I,J,B=1,\cdots , \mathfrak{D}\\ I <J}}{\max } \{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1\). Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with \(\mathfrak{D}= 3\) and \(\widetilde{q}_1 \approx \widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3\), which corresponds to the stability of the Friedmann-Lemaître-Robertson-Walker solution's Big Bang or (2) the Einstein-vacuum equations for \(\mathfrak{D}\geq 38\) with \(\underset{I=1,\cdots ,\mathfrak{D}}{\max } |\widetilde{q}_I| < 1/6\). In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the \(1+\mathfrak{D} \)-dimensional Einstein-scalar field system for all \(\mathfrak{D}\geq 3\) and the \(1+\mathfrak{D} \)-dimensional Einstein-vacuum equations for \(\mathfrak{D}\geq 10\); both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in \(1+3\) dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized \(U(1)\)-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized \(U(1)\)-symmetric solutions.
Our proof relies on a new formulation of Einstein's equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi-Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to \(t\), and to handle this difficulty, we use \(t\)-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the \(t\)-weights and interpolating between the singularity-strength of the solution's low order and high order derivatives. Finally, we note that our formulation of Einstein's equations highlights the quantities that might generate instabilities outside of the sub-critical regime.Continuous spectrum on cosmological colliderhttps://zbmath.org/1522.832722023-12-07T16:00:11.105023Z"Aoki, Shuntaro"https://zbmath.org/authors/?q=ai:aoki.shuntaro(no abstract)Inflation in metric-affine quadratic gravityhttps://zbmath.org/1522.832852023-12-07T16:00:11.105023Z"D. Gialamas, Ioannis"https://zbmath.org/authors/?q=ai:gialamas.ioannis-d"Tamvakis, Kyriakos"https://zbmath.org/authors/?q=ai:tamvakis.kyriakos(no abstract)Tabletop potentials for inflation from \(f(R)\) gravityhttps://zbmath.org/1522.833072023-12-07T16:00:11.105023Z"Shtanov, Yuri"https://zbmath.org/authors/?q=ai:shtanov.yuri"Sahni, Varun"https://zbmath.org/authors/?q=ai:sahni.varun"Mishra, Swagat S."https://zbmath.org/authors/?q=ai:mishra.swagat-s(no abstract)The geometry and topology of stationary multiaxisymmetric vacuum black holes in higher dimensionshttps://zbmath.org/1522.833122023-12-07T16:00:11.105023Z"Kakkat, Vishnu"https://zbmath.org/authors/?q=ai:kakkat.vishnu"Khuri, Marcus"https://zbmath.org/authors/?q=ai:khuri.marcus-a"Rainone, Jordan"https://zbmath.org/authors/?q=ai:rainone.jordan"Weinstein, Gilbert"https://zbmath.org/authors/?q=ai:weinstein.gilbertThe authors extend their previous work to 5 dimensions and prove the existence and uniqueness of solutions of the reduced Einstein equations for vacuum black holes in \(n(n+3)\)-dimensional spacetimes admitting the isometry group \(\mathbb{R}\times U(1)^n\), with Kaluza-Klein asymmetries for \(n\geq 3\). They also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in lower dimensions.
Furthermore they provide a counterexample to a conjecture of \textit{S. Hollands} and \textit{A. Ishibashi} [Classical Quantum Gravity 29, No. 16, Article ID 163001, 47 p. (2012; Zbl 1252.83005)] concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.
Reviewer: Alex B. Gaina (Chişinău)The supersymmetric Neveu-Schwarz branes of non-relativistic string theoryhttps://zbmath.org/1522.833202023-12-07T16:00:11.105023Z"Bergshoeff, E. A."https://zbmath.org/authors/?q=ai:bergshoeff.eric-a"Lahnsteiner, J."https://zbmath.org/authors/?q=ai:lahnsteiner.johannes"Romano, L."https://zbmath.org/authors/?q=ai:romano.luca"Rosseel, J."https://zbmath.org/authors/?q=ai:rosseel.janSummary: We construct the basic Neveu-Schwarz (NS) brane solutions of non-relativistic string theory using longitudinal T-duality as a solution generating technique. Extending the NS background fields to a supergravity multiplet, we verify that all solutions we find are half-supersymmetric. The two perturbative solutions we find both have an interpretation as the background geometry outside a string-like object. Correspondingly, we refer to these non-Lorentzian backgrounds as winding string and unwound string solution. Whereas the winding string is part of the on-shell spectrum of non-relativistic string theory, the unwound string only makes sense off-shell where it mediates the instantaneous gravitational force. Seen from the nine-dimensional point of view, we find that the winding string solution is sourced by a non-relativistic massive particle and that the unwound string solution is sourced by a massless Galilean particle of zero colour and spin. We explain how these two string solutions fit into a discrete lightcone quantization of string theory. We shortly discuss the basic NS five-brane and Kaluza-Klein monopole solutions and show that they are both half-supersymmetric.Torsional string Newton-Cartan geometry for non-relativistic stringshttps://zbmath.org/1522.833212023-12-07T16:00:11.105023Z"Bidussi, Leo"https://zbmath.org/authors/?q=ai:bidussi.leo"Harmark, Troels"https://zbmath.org/authors/?q=ai:harmark.troels"Hartong, Jelle"https://zbmath.org/authors/?q=ai:hartong.jelle"Obers, Niels A."https://zbmath.org/authors/?q=ai:obers.niels-a"Oling, Gerben"https://zbmath.org/authors/?q=ai:oling.gerben-w-jSummary: We revisit the formulation of non-relativistic (NR) string theory and its target space geometry. We obtain a new formulation in which the geometry contains a two-form field that couples to the tension current and that transforms under string Galilei boosts. This parallels the Newton-Cartan one-form that couples to the mass current of a non-relativistic point particle. We show how this formulation of the NR string arises both from an infinite speed of light limit and a null reduction of the relativistic closed bosonic string. In both cases, the two-form originates from a combination of metric quantities and the Kalb-Ramond field. The target space geometry of the NR string is seen to arise from the gauging of a new algebra that is obtained by an İnönü-Wigner contraction of the Poincaré algebra extended by the symmetries of the Kalb-Ramond field. In this new formulation, there are no superfluous target space fields that can be removed by fixing a Stückelberg symmetry. Classically, there are no foliation/torsion constraints imposed on the target space geometry.Duality invariant string beta functions at two loopshttps://zbmath.org/1522.833222023-12-07T16:00:11.105023Z"Bonezzi, Roberto"https://zbmath.org/authors/?q=ai:bonezzi.roberto"Codina, Tomas"https://zbmath.org/authors/?q=ai:codina.tomas"Hohm, Olaf"https://zbmath.org/authors/?q=ai:hohm.olafSummary: We compute, for cosmological backgrounds, the \(O(d, d; \mathbb{R})\) invariant beta functions for the sigma model of the bosonic string at two loops. This yields an independent first-principle derivation of the order \(\alpha^\prime\) corrections to the cosmological target-space equations. To this end we revisit the quantum consistency of Tseytlin's duality invariant formulation of the worldsheet theory. While we confirm the absence of gravitational (and hence Lorentz) anomalies, our results show that the minimal subtraction scheme is not applicable, implying significant technical complications at higher loops. To circumvent these we then change gears and use the Polyakov action for cosmological backgrounds, applying a suitable perturbation scheme that, although not \(O(d, d; \mathbb{R})\) invariant, allows one to efficiently determine the \(O(d, d; \mathbb{R})\) invariant beta functions.BPS surface operators and calibrationshttps://zbmath.org/1522.833322023-12-07T16:00:11.105023Z"Drukker, Nadav"https://zbmath.org/authors/?q=ai:drukker.nadav"Trépanier, Maxime"https://zbmath.org/authors/?q=ai:trepanier.maximeSummary: We present here a careful study of the holographic duals of BPS surface operators in the 6d \(\mathcal{N} = (2, 0)\) theory. Several different classes of surface operators have been recently identified and each class has a specific calibration form -- a 3-form in \(AdS_7\times S^4\) whose pullback to the M2-brane world-volume is equal to the volume form. In all but one class, the appropriate forms are exact, so the action of the M2-brane is easily expressed in terms of boundary data, which is the geometry of the surface. Specifically, for surfaces of vanishing anomaly, it is proportional to the integral of the square of the extrinsic curvature.
This can be extended to the case of surfaces with anomalies, by taking the ratio of two surfaces with the same anomaly. This gives a slew of new expectation values at large \(N\) in this theory. For one specific class of surface operators, which are Lagrangian submanifolds of \(\mathbb{R}^4 \subset \mathbb{R}^6\), the structure is far richer and we find that the M2-branes are special Lagrangian submanifold of an appropriate six-dimensional almost Calabi-Yau submanifold of \(AdS_7\times S^4\). This allows for an elegant treatment of many such examples.Coset space actions for nonrelativistic stringshttps://zbmath.org/1522.833342023-12-07T16:00:11.105023Z"Fontanella, Andrea"https://zbmath.org/authors/?q=ai:fontanella.andrea"van Tongeren, Stijn J."https://zbmath.org/authors/?q=ai:van-tongeren.stijn-jSummary: We formulate the stringy nonrelativistic limits of the flat space and \(\mathrm{AdS}_5\times\mathrm{S}^5\) string as coset models, based on the string Bargmann and extended string Newton-Hooke algebras respectively. Our construction mimics the typical relativistic one, but differs in several interesting ways. Using our coset formulation we give a Lax representation of the equations of motion of both models.Intrinsic quantum dynamics of particles in brane gravityhttps://zbmath.org/1522.833362023-12-07T16:00:11.105023Z"Jalalzadeh, Shahram"https://zbmath.org/authors/?q=ai:jalalzadeh.shahramSummary: The Newtonian dynamics of particles in brane gravity is investigated. Due to the coupling of the particles' energy-momentum tensor to the tension of the brane, the particle is semi-confined and oscillates along the extra dimension. We demonstrate that the frequency of these oscillations is proportional to the kinetic energy of the particle in the brane. We show that the classical stability of particle trajectories on the brane gives us the Bohr-Sommerfeld quantization condition. The particle's motion along the extra dimension allows us to formulate a geometrical version of the uncertainty principle. Furthermore, we exhibited that the particle's motion along the extra dimension is identical to the time-independent Schrödinger equation. The dynamics of a free particle, particles in a box, a harmonic oscillator, a bouncing particle, and tunneling are re-examined. We show that the particle's motion along the extra dimension yields a quantized energy spectrum for bound states.LVS de Sitter vacua are probably in the swamplandhttps://zbmath.org/1522.833372023-12-07T16:00:11.105023Z"Junghans, Daniel"https://zbmath.org/authors/?q=ai:junghans.danielSummary: We argue that dS vacua in the LARGE-volume scenario of type IIB string theory are vulnerable to various unsuppressed curvature, warping and \(g_s\) corrections. We work out in general how these corrections affect the moduli vevs, the vacuum energy and the moduli masses in the 4D EFT for the two Kähler moduli, the conifold modulus and a nilpotent superfield describing the anti-brane uplift. Our analysis reveals that the corrections are parametrically larger in the relevant expressions than one might have guessed from their suppression in the off-shell potential. Some corrections appear without any parametric suppression at all, which makes them particularly dangerous for candidate dS vacua. Other types of corrections can in principle be made small for appropriate parameter choices. However, we show in an explicit model that this is never possible for all corrections at the same time when the vacuum energy is positive. Some of the corrections we consider are also relevant for the stability of non-supersymmetric AdS vacua.New formulation of non-relativistic string in \(AdS_5 \times S^5\)https://zbmath.org/1522.833382023-12-07T16:00:11.105023Z"Klusoň, J."https://zbmath.org/authors/?q=ai:kluson.josefSummary: We study non-relativistic limit of \(AdS_5 \times S^5\) background and determine corresponding Newton-Cartan fields. We also find canonical form of this new formulation of non-relativistic string in this background and discuss its formulation in the uniform light-cone gauge.Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signaturehttps://zbmath.org/1522.833442023-12-07T16:00:11.105023Z"Médevielle, M."https://zbmath.org/authors/?q=ai:medevielle.maxime"Mohaupt, T."https://zbmath.org/authors/?q=ai:mohaupt.thomas"Pope, G."https://zbmath.org/authors/?q=ai:pope.giacomo|pope.graeme|pope.gary-aSummary: We obtain the bosonic Lagrangians of vector and hypermultiplets coupled to four-dimensional \(\mathcal{N} = 2\) supergravity in signatures \((0, 4)\), \((1, 3)\) and \((2, 2)\) by compactification of type-II string theories in signatures \((0, 10)\), \((1, 9)\) and \((2, 8)\) on a Calabi-Yau threefold. Depending on the signature and the distinctions between type-IIA/\(\mathrm{IIA}^\ast\)/IIB/\(\mathrm{IIB}^\ast\)/IIB' the resulting scalar geometries are special Kähler or special para-Kähler for vector multiplets and quaternion-Kähler or para-quaternion Kähler for hypermultiplets. By spacelike and timelike reductions we obtain three-dimensional \(\mathcal{N} = 4\) supergravity theories coupled to two sets of hypermultiplets. We determine the c-maps relating vector to hypermultiplets, and show how the four-dimensional theories are related by spacelike, timelike and mixed, signature-changing T-dualities.Exact scalar (quasi-)normal modes of black holes and solitons in gauged SUGRAhttps://zbmath.org/1522.833572023-12-07T16:00:11.105023Z"Aguayo, Monserrat"https://zbmath.org/authors/?q=ai:aguayo.monserrat"Hernández, Ankai"https://zbmath.org/authors/?q=ai:hernandez.ankai"Mena, José"https://zbmath.org/authors/?q=ai:mena.jose"Oliva, Julio"https://zbmath.org/authors/?q=ai:oliva.julio"Oyarzo, Marcelo"https://zbmath.org/authors/?q=ai:oyarzo.marceloSummary: In this paper we identify a new family of black holes and solitons that lead to the exact integration of scalar probes, even in the presence of a non-minimal coupling with the Ricci scalar which has a non-trivial profile. The backgrounds are planar and spherical black holes as well as solitons of \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) \(\mathcal{N} = 4\) gauged supergravity in four dimensions. On these geometries, we compute the spectrum of (quasi-)normal modes for the non-minimally coupled scalar field. We find that the equation for the radial dependence can be integrated in terms of hypergeometric functions leading to an exact expression for the frequencies. The solutions do not asymptote to a constant curvature spacetime, nevertheless the asymptotic region acquires an extra conformal Killing vector. For the black hole, the scalar probe is purely ingoing at the horizon, and requiring that the solutions lead to an extremum of the action principle we impose a Dirichlet boundary condition at infinity. Surprisingly, the quasinormal modes do not depend on the radius of the black hole, therefore this family of geometries can be interpreted as isospectral in what regards to the wave operator non-minimally coupled to the Ricci scalar. We find both purely damped modes, as well as exponentially growing unstable modes depending on the values of the non-minimal coupling parameter. For the solitons we show that the same integrability property is achieved separately in a non-supersymmetric solutions as well as for the supersymmetric one. Imposing regularity at the origin and a well defined extremum for the action principle we obtain the spectra that can also lead to purely oscillatory modes as well as to unstable scalar probes, depending on the values of the non-minimal coupling.Massive flows in \(\mathrm{AdS}_6/\mathrm{CFT}_5\)https://zbmath.org/1522.833582023-12-07T16:00:11.105023Z"Akhond, Mohammad"https://zbmath.org/authors/?q=ai:akhond.mohammad"Legramandi, Andrea"https://zbmath.org/authors/?q=ai:legramandi.andrea"Nunez, Carlos"https://zbmath.org/authors/?q=ai:nunez.carlos"Santilli, Leonardo"https://zbmath.org/authors/?q=ai:santilli.leonardo"Schepers, Lucas"https://zbmath.org/authors/?q=ai:schepers.lucasSummary: We study five-dimensional \(\mathcal{N} = 1\) Superconformal Field Theories of the linear quiver type. These are deformed by a relevant operator, corresponding to a homogeneous mass term for certain matter fields. The free energy is calculated at arbitrary values of the mass parameter. After a careful regularisation procedure, the result can be put in correspondence with a calculation in the supergravity dual background. The F-theorem is verified for these flows, both in field theory and in supergravity. This letter presents some of the results in the companion paper [\textit{M. Akhond} et al., ``Matrix models and holography: mass deformations of long quiver theories in 5d and 3d'', Preprint, \url{arXiv:2211.13240}].TCFHs, hidden symmetries and type II theorieshttps://zbmath.org/1522.833862023-12-07T16:00:11.105023Z"Grimanellis, L."https://zbmath.org/authors/?q=ai:grimanellis.l"Papadopoulos, G."https://zbmath.org/authors/?q=ai:papadopoulos.george.1|papadopoulos.george-k|papadopoulos.g-j|papadopoulos.george-d|papadopoulos.gregory|papadopoulos.g-th|papadopoulos.g-j.1|papadopoulos.georgios-o|papadopoulos.george|papadopoulos.george-a|papadopoulos.george-a.1|papadopoulos.george-k.1"Phillips, J."https://zbmath.org/authors/?q=ai:phillips.john-w.1|phillips.jeff-m|phillips.john|phillips.joel-r|phillips.jon-d|phillips.jack-raymond|phillips.j-richard|phillips.jeffrey-c|phillips.james-d|phillips.j-p-n|phillips.james-l|phillips.jason-d|phillips.jerry-l|phillips.j-a.1|phillips.jim|phillips.john-w.2|phillips.james-g|phillips.james-c.1|phillips.jonathan|phillips.james-r|phillips.james-w|phillips.jeremy-c|phillips.john-f|phillips.jeffrey-mark|phillips.james-b|phillips.j-h|phillips.joseph|phillips.jen|phillips.p-jonathon|phillips.james-mSummary: We present the twisted covariant form hierarchies (TCFH) of type IIA and IIB 10-dimensional supergravities and show that all form bilinears of supersymmetric backgrounds satisfy the conformal Killing-Yano equation with respect to a TCFH connection. We also compute the Killing-Stäckel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric type II brane backgrounds and demonstrate that the geodesic flow on these solutions is completely integrable by giving all independent charges in involution. We then identify all form bilinears of common sector and D-brane backgrounds which generate hidden symmetries for particle and string probe actions. We also explore the question on whether charges constructed from form bilinears are sufficient to prove the integrability of probes on supersymmetric backgrounds.Erratum to: ``Probing non-Gaussianities with the high frequency tail of induced gravitational waves''https://zbmath.org/1522.834052023-12-07T16:00:11.105023Z"Atal, Vicente"https://zbmath.org/authors/?q=ai:atal.vicente"Domènech, Guillem"https://zbmath.org/authors/?q=ai:domenech.guillemErratum to the authors' paper [ibid. 2021, No. 6, Paper No. 1, 37 p. (2021; Zbl 1485.83114)].Boson mixing and flavor vacuum in the expanding universe: a possible candidate for the dark energyhttps://zbmath.org/1522.834152023-12-07T16:00:11.105023Z"Capolupo, Antonio"https://zbmath.org/authors/?q=ai:capolupo.antonio"Quaranta, Aniello"https://zbmath.org/authors/?q=ai:quaranta.anielloSummary: We analyze the boson mixing in curved spacetime and compute the expectation value of the energy-momentum tensor of bosons on the flavor vacuum in spatially flat Friedmann-Lemaître-Robertson-Walker metrics. We show that the energy-momentum tensor of the flavor vacuum behaves as the effective energy-momentum tensor of a perfect fluid. Assuming a fixed de Sitter background, we show that the equation of state can assume values in the interval \([-1, 1]\), and, in the flat space-time limit has a value \(-1\), which is the one of the dark energy. The results here presented show that vacuum of mixed bosons like neutrino super-partners, can represent a possible component of the dark energy of the Universe.Ghost and Laplacian instabilities in teleparallel Horndeski gravityhttps://zbmath.org/1522.834162023-12-07T16:00:11.105023Z"Capozziello, Salvatore"https://zbmath.org/authors/?q=ai:capozziello.salvatore"Caruana, Maria"https://zbmath.org/authors/?q=ai:caruana.maria"Levi Said, Jackson"https://zbmath.org/authors/?q=ai:said.jackson-levi"Sultana, Joseph"https://zbmath.org/authors/?q=ai:sultana.joseph(no abstract)On the viability of \(f(Q)\) gravity modelshttps://zbmath.org/1522.834182023-12-07T16:00:11.105023Z"De, Avik"https://zbmath.org/authors/?q=ai:de.avik"Loo, Tee-How"https://zbmath.org/authors/?q=ai:loo.tee-howSummary: In general relativity, the contracted Bianchi identity makes the field equation compatible with the energy conservation, likewise in \(f(R)\) theories of gravity. We show that this classical phenomenon is not guaranteed in the symmetric teleparallel theory, and rather generally \(f(Q)\) model specific. We further prove that the energy conservation criterion is equivalent to the affine connection's field equation of \(f(Q)\) theory, and except the \(f(Q) = \alpha Q + \beta\) model, the non-linear \(f(Q)\) models do not satisfy the energy conservation or, equivalently the second field equation in every spacetime geometry; unless \(Q\) itself is a constant. So the problem is deep-rooted in the theory, several physically motivated examples are provided in the support.Global dynamics for a collisionless charged plasma in Bianchi spacetimes in Eddington-inspired-Born-Infeld gravityhttps://zbmath.org/1522.834202023-12-07T16:00:11.105023Z"Djiodjo Seugmo, Guichard"https://zbmath.org/authors/?q=ai:djiodjo-seugmo.guichard"Tadmon, Calvin"https://zbmath.org/authors/?q=ai:tadmon.calvinSummary: We consider a Bianchi type I--IX physical metric \(g\), an auxiliary metric \(q\) with a collisionless charged relativistic plasma in Eddington-inspired-Born-Infeld theory. We first derive a governing system of second order nonlinear partial differential equations. Then, by the characteristics method applied to the Vlasov equation whose solution is the distribution function \(f\), we manage to construct an iterated sequence. This leads to the existence and uniqueness of a local solution on an interval \([0, T)\), with \(T < \infty\). Then under certain assumptions of smallness on the mean curvatures in both directions, the Eddington parameter \(k\) and the dimensionless parameter \(\lambda\), we show that this solution is global in time.Ultraviolet-regularized power spectrum without infrared distortions in cosmological spacetimeshttps://zbmath.org/1522.834232023-12-07T16:00:11.105023Z"Ferreiro, Antonio"https://zbmath.org/authors/?q=ai:ferreiro.antonio"Torrenti, Francisco"https://zbmath.org/authors/?q=ai:torrenti.franciscoSummary: We reexamine the regularization of the two-point function of a scalar field in a Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. Adiabatic regularization provides a set of subtraction terms in momentum space that successfully remove its ultraviolet divergences at coincident points, but can significantly distort the power spectrum at infrared scales, especially for light fields. In this work we propose, by using the intrinsic ambiguities of the renormalization program, a new set of subtraction terms that minimize the distortions for scales \(k \lesssim M\), with \(M\) an arbitrary mass scale. Our method is consistent with local covariance and equivalent to general regularization methods in curved spacetime. We apply our results to the regularization of the power spectrum in de Sitter space: while the adiabatic scheme yields exactly \(\Delta_\phi^{(\mathrm{reg})} = 0\) for a massless field, our proposed prescription recovers the standard scale-invariant result \(\Delta_\phi^{(\mathrm{reg})} \simeq H^2/(4\pi^2)\) at super-horizon scales.A canonical complex structure and the bosonic signature operator for scalar fields in globally hyperbolic spacetimeshttps://zbmath.org/1522.834242023-12-07T16:00:11.105023Z"Finster, Felix"https://zbmath.org/authors/?q=ai:finster.felix"Much, Albert"https://zbmath.org/authors/?q=ai:much.albertSummary: The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.Quantum cosmology of pure connection general relativityhttps://zbmath.org/1522.834272023-12-07T16:00:11.105023Z"Gielen, Steffen"https://zbmath.org/authors/?q=ai:gielen.steffen"Nash, Elliot"https://zbmath.org/authors/?q=ai:nash.elliotSummary: We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebański formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.Chirally factorised truncated conformal space approachhttps://zbmath.org/1522.834342023-12-07T16:00:11.105023Z"Horváth, D. X."https://zbmath.org/authors/?q=ai:horvath.david-x"Hódsági, K."https://zbmath.org/authors/?q=ai:hodsagi.kristof"Takács, G."https://zbmath.org/authors/?q=ai:takacs.gaborSummary: Truncated Conformal Space Approach (TCSA) is a highly efficient method to compute spectra, operator matrix elements and time evolution in quantum field theories defined as relevant perturbations of \(1 + 1\)-dimensional conformal field theories. However, similarly to other exact diagonalisation methods, TCSA is ridden with the ``curse of dimensionality'': the dimension of the Hilbert space increases exponentially with the (square root of the) truncation level, limiting its precision by the available memory resources. Here we describe an algorithm which exploits the chiral factorisation property of conformal field theory with periodic boundary conditions to achieve a substantial improvement in the truncation level. The Chirally Factorised TCSA (CFTCSA) algorithm presented here works with inputs describing the necessary CFT data in a specified format. It makes possible much more precise calculations with given computing resources and extends the reach of the method to problems requiring large Hilbert space dimensions. In fact, it has already been used in a number of recent works ranging from determination of form factors, through studying confinement of topological excitations to non-equilibrium dynamics. Besides the description of the algorithm, a MATLAB implementation of the algorithm is also provided as an ancillary file package, supplemented with example codes computing spectra, matrix elements and time evolution, and with CFT data for three different quantum field theories. We also give a detailed how-to guide for constructing the required CFT data for Vir We also give a detailed how-to guide for constructing the required CFT data for Viasoro minimal models with central charge \(c < 1\), and for the massless free boson with \(c = 1\).Uses of complex metrics in cosmologyhttps://zbmath.org/1522.834372023-12-07T16:00:11.105023Z"Jonas, Caroline"https://zbmath.org/authors/?q=ai:jonas.caroline"Lehners, Jean-Luc"https://zbmath.org/authors/?q=ai:lehners.jean-luc"Quintin, Jerome"https://zbmath.org/authors/?q=ai:quintin.jeromeSummary: Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more probable than ordinary classical evolution. In the cosmological context, we investigate a criterion proposed by Witten (based on works of \textit{M. Kontsevich} and \textit{G. Segal} [Q. J. Math. 72, No. 1--2, 673--699 (2021; Zbl 1471.81075)] and of \textit{J. Louko} and \textit{R. D. Sorkin} [Classical Quantum Gravity 14, No. 1, 179--203 (1997; Zbl 0868.53069)]) to decide whether a complex metric is allowable or not. Because of the freedom to deform complex metrics using Cauchy's theorem, deciding whether a metric is allowable in general requires solving a complicated optimisation problem. We describe a method that allows one to quickly determine the allowability of minisuperspace metrics. This enables us to study the off-shell structure of minisuperspace path integrals, which we investigate for various boundary conditions. Classical transitions always reside on the boundary of the domain of allowable metrics, and care must be taken in defining appropriate integration contours for the corresponding gravitational path integral. Perhaps more surprisingly, we find that proposed quantum (`tunnelling') transitions from a contracting to an expanding universe violate the allowability criterion and may thus be unphysical. No-boundary solutions, by contrast, are found to be allowable, and moreover we demonstrate that with an initial momentum condition an integration contour over allowable metrics may be explicitly described in arbitrary spacetime dimensions.Holographic characterisation of locally anti-de Sitter spacetimeshttps://zbmath.org/1522.834472023-12-07T16:00:11.105023Z"McGill, Alex"https://zbmath.org/authors/?q=ai:mcgill.alexSummary: It is shown that an \((n+1)\)-dimensional asymptotically anti-de Sitter solution of the Einstein-vacuum equations is locally isometric to pure anti-de Sitter spacetime near a region of the conformal boundary if and only if the boundary metric is conformally flat and (for \(n \neq 4\)) the boundary stress-energy tensor vanishes, subject to (i) sufficient (finite) regularity in the metric and (ii) the satisfaction of a conformally invariant geometric criterion on the boundary region. A key tool in the proof is the Carleman estimate of \textit{A. Chatzikaleas} and \textit{A. Shao} [Commun. Math. Phys. 395, No. 2, 521--570 (2022; Zbl 1510.83026)] -- a generalisation of previous work by \textit{A. McGill} and \textit{A. Shao} in [Classical Quantum Gravity 38, No. 5, Article ID 054001, 63 p. (2021; Zbl 1480.83030)] -- which is applied to prove a unique continuation result for the Weyl curvature at the conformal boundary given vanishing to sufficiently high order over the boundary region.General effective field theory of teleparallel gravityhttps://zbmath.org/1522.834492023-12-07T16:00:11.105023Z"Mylova, Maria"https://zbmath.org/authors/?q=ai:mylova.maria"Said, Jackson Levi"https://zbmath.org/authors/?q=ai:said.jackson-levi"Saridakis, Emmanuel N."https://zbmath.org/authors/?q=ai:saridakis.emmanuel-nSummary: We construct the effective field theory (EFT) of the teleparallel equivalent of general relativity (TEGR). Firstly, we present the necessary field redefinitions of the scalar field and the tetrads. Then we provide all the terms at next-to-leading-order, containing the torsion tensor and its derivatives, and derivatives of the scalar field, accompanied by generic scalar-field-dependent couplings, where all operators are suppressed by a scale \(\Lambda\). Removing all redundant terms using the field redefinitions we result to the EFT of TEGR, which includes significantly more terms comparing to the EFT of general relativity (GR). Finally, we present an application in a cosmological framework. Interestingly enough, although GR and TEGR are completely equivalent at the level of classical equations, we find that their corresponding EFTs possess minor but non-zero differences. Hence, we do verify that at higher energies the excitation and the features of the extra degrees of freedom are slightly different in the two theories, thus making them theoretically distinguishable. Nevertheless, we mention that these differences are suppressed by the heavy mass scale \(\Lambda\) and thus it is not guaranteed that they could be measured in future experiments and observations.Gravitational chiral anomaly and anomalous transport for fields with spin 3/2https://zbmath.org/1522.834512023-12-07T16:00:11.105023Z"Prokhorov, G. Yu."https://zbmath.org/authors/?q=ai:prokhorov.g-yu"Teryaev, O. V."https://zbmath.org/authors/?q=ai:teryaev.oleg-v"Zakharov, V. I."https://zbmath.org/authors/?q=ai:zakharov.valentin-iSummary: In a fluid with vorticity and acceleration, an axial current arises in the third order of gradient expansion, the corresponding phenomenon can be called the Kinematical vortical effect (KVE). It has recently been shown that the KVE while existing in the absence of gravitational fields, is nonetheless associated with effects in curved space-time, namely with the gravitational chiral quantum anomaly. In this paper, we find the KVE transport coefficients using the Zubarev quantum-statistical density operator for the Rarita-Schwinger-Adler theory, which includes massless fields with spins 3/2 and 1/2, and demonstrate the relationship with the gravitational anomaly. A prediction is made about the possible form of the transport coefficients for massless fields with arbitrary spin.Anomaly-free scale symmetry and gravityhttps://zbmath.org/1522.834532023-12-07T16:00:11.105023Z"Shaposhnikov, Mikhail"https://zbmath.org/authors/?q=ai:shaposhnikov.mikhail"Tokareva, Anna"https://zbmath.org/authors/?q=ai:tokareva.annaSummary: In this Letter, we address the question of whether the conformal invariance can be considered as a global symmetry of a theory of fundamental interactions. To describe Nature, this theory must contain a mechanism of spontaneous breaking of the scale symmetry. Besides that, the fundamental theory must include gravity, whereas all known extensions of the conformal invariance to the curved space-time suffer from the Weyl anomaly. We show that conformal symmetry can be made free from quantum anomaly only in flat space. The presence of gravity would reduce the global symmetry group of the fundamental theory to the scale invariance only. We discuss how the effective Lagrangian respecting the scale symmetry can be used for the description of particle phenomenology and cosmology.Inflationary Bianchi type-II spacetime with exponential potentialhttps://zbmath.org/1522.834572023-12-07T16:00:11.105023Z"Tinker, Seema"https://zbmath.org/authors/?q=ai:tinker.seemaSummary: In the present study, I have calculated LRS Bianchi Type II inflationary spacetime under the influence of effective potential \(V(\phi) = \exp(-\lambda\phi)\), \(\phi > 0\), where \(\phi\) represents Higg's field. To obtain the precise solution average scale factor \(\widetilde{a}(t)\) is advised as \(\widetilde{a}^3 = (R^2)S=\exp([\lambda\phi(t)]\). The model ee becomes isotropic and shear-free in a special case. Also, the model exhibits no singularity at the initial stage. I have discussed the physical and kinematic behavior of the model using some dynamical parameters.Neutron stars in modified teleparallel gravityhttps://zbmath.org/1522.850072023-12-07T16:00:11.105023Z"Vilhena, S. G."https://zbmath.org/authors/?q=ai:vilhena.s-g"Duarte, S. B."https://zbmath.org/authors/?q=ai:duarte.sergio-b"Dutra, M."https://zbmath.org/authors/?q=ai:dutra.max-s|dutra.maira"Pompeia, P. J."https://zbmath.org/authors/?q=ai:pompeia.pedro-jose(no abstract)On the convex Pfaff-Darboux theorem of Ekeland and Nirenberghttps://zbmath.org/1522.911212023-12-07T16:00:11.105023Z"Bryant, Robert L."https://zbmath.org/authors/?q=ai:bryant.robert-lSummary: The classical Pfaff-Darboux theorem, which provides local `normal forms' for \(1\)-forms on manifolds, has applications in the theory of certain economic models [\textit{P. A. Chiappori} and \textit{I. Ekeland}, Found. Trends Microecon. 5, No. 1--2, 1--151 (2009; Zbl 1192.91004)]. However, the normal forms needed in these models often come with an additional requirement of some type of convexity, which is not provided by the classical proofs of the Pfaff-Darboux theorem. (The appropriate notion of `convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in \(\mathbb{R}^n\), convexity has its usual meaning.) In [Methods Appl. Anal. 9, No. 3, 329--344 (2002; Zbl 1082.58501)], \textit{I. Ekeland} and \textit{L. Nirenberg} were able to characterize necessary and sufficient conditions for a given \(1\)-form \(\omega\) to admit a convex local normal form (and to show that some earlier attempts \textit{P.-A. Chiappori} and \textit{I. Ekeland} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, No. 1--2, 287--297 (1997; Zbl 1004.58501)] and [\textit{V. M. Zakalyukin}, C. R. Acad. Sci., Paris, Sér. I, Math. 327, No. 7, 633--638 (1998; Zbl 1004.58502)] at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result of Ekeland and Nirenberg concerns the case in which the affine connection is flat).