Recent zbMATH articles in MSC 53Chttps://zbmath.org/atom/cc/53C2023-11-13T18:48:18.785376ZWerkzeugDetermining triangulations and quadrangulations by boundary distanceshttps://zbmath.org/1521.050342023-11-13T18:48:18.785376Z"Haslegrave, John"https://zbmath.org/authors/?q=ai:haslegrave.johnSummary: We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. However, we show that a natural conjecture for a ``mixed'' version of the two results is not true.Curvatures, graph products and Ricci flatnesshttps://zbmath.org/1521.051702023-11-13T18:48:18.785376Z"Cushing, David"https://zbmath.org/authors/?q=ai:cushing.david"Kamtue, Supanat"https://zbmath.org/authors/?q=ai:kamtue.supanat"Kangaslampi, Riikka"https://zbmath.org/authors/?q=ai:kangaslampi.riikka"Liu, Shiping"https://zbmath.org/authors/?q=ai:liu.shiping"Peyerimhoff, Norbert"https://zbmath.org/authors/?q=ai:peyerimhoff.norbertSummary: In this paper, we compare Ollivier-Ricci curvature and Bakry-Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier-Ricci curvature implies the nonnegativity of Bakry-Émery curvature under triangle-freeness and an additional in-degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While nonnegativity of both curvatures is preserved under Cartesian products, we show that in the case of strong products, nonnegativity of Ollivier-Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance-regular graphs of girth 4 attain their maximal possible curvature values.Eisenstein metricshttps://zbmath.org/1521.110222023-11-13T18:48:18.785376Z"Franc, Cameron"https://zbmath.org/authors/?q=ai:franc.cameronThe Eisenstein series, which are named after German mathematician Gotthold Eisenstein (16 April 1823--11 October 1852), are particular modular forms with infinite series expansions. On the modular group, the Eisenstein series can be to automorphic forms, which are well-behaved functions from a topological group. The Eisenstein series form the most explicit examples of modular forms for the full modular group \(\mathrm{SL}(2,\mathbb Z)\). Since the space of modular forms of weight \(2k\) has dimension 1 for \(2k = 4, 6, 8, 10, 14\), different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In the theory of Eisenstein series it is well known that a simple function satisfying a linear differential equation and averages to get a more interesting function satisfying a larger set of invariance properties. By linearity, the averaged function satisfies the same linear differential equation.
The content of the paper covers Kloosterman sums, Eisenstein series, harmonic metric, modular group, Eisenstein metrics, tame harmonic metrics, the inclusion representation, unitary monodromy at the cusp, Higgs field among others. The author studies the Eisenstein metrics \(H(\tau, s)\) with representations of \(\mathrm{SL}(2,\mathbb Z)\) generalizing previous constructions. The author provides the convergence related to the real part. The author gives many results and relations among Eisenstein series, modular group, Eisenstein metrics, harmonic metric, tame harmonic metrics with comments and examples.
Reviewer: Yilmaz Simsek (Antalya)Constant scalar curvature Kähler metrics on rational surfaceshttps://zbmath.org/1521.140302023-11-13T18:48:18.785376Z"Martinez-Garcia, Jesus"https://zbmath.org/authors/?q=ai:martinez-garcia.jesusSummary: We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rational surfaces other than those two admit a destabilising slope test configuration for some polarisation, as introduced by Ross and Thomas. We further show that all Hirzebruch surfaces other than the quadric surface and all rational surfaces with Picard rank 3 do not admit a constant scalar curvature Kähler metric in any Kähler class.
{{\copyright} 2021 Wiley-VCH GmbH}Domination results in \(n\)-Fuchsian fibers in the moduli space of Higgs bundleshttps://zbmath.org/1521.140642023-11-13T18:48:18.785376Z"Dai, Song"https://zbmath.org/authors/?q=ai:dai.song"Li, Qiongling"https://zbmath.org/authors/?q=ai:li.qionglingThe aim of this paper is to show some domination results in \(n\)-Fuchsian fibers in the moduli space of Higgs bundles. The authors show that the energy density of the associated harmonic map of an \(n\)-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, they obtain the existence and uniqueness of equivariant minimal or maximal surfaces in a certain product Riemannian or pseudo-Riemannian manifold. Their proof is based on establishing an algebraic inequality generalizing a theorem of Ness on the nilpotent orbits to general orbits.
This paper is organized as follows: Section 1 is an introduction to the subject and statement of the main result. In Section 2 the authors introduce some facts about Ness' theorem [\textit{L. Ness}, Am. J. Math. 106, 1281--1329 (1984; Zbl 0604.14006)] on the nilpotent orbits generalize this result to the general case. This result plays a key role in this paper and the proof will be postponed to Section 3. Section 4 deals with domination results. In this section, the authors first recall some preliminaries in the non-abelian Hodge theory and higher Teichmüller theory and prove the main theorems. This whole section 5 is devoted to proving a proposition, which plays an important role in the proof of the theorems in the previous section. In Section 6, the authors show some applications of the domination results. They derive two main applications from Sections 4 and 5 to equivariant minimal surfaces and maximal surfaces in product spaces.
Reviewer: Ahmed Lesfari (El Jadida)Delta-invariants of complete intersection log del Pezzo surfaceshttps://zbmath.org/1521.140662023-11-13T18:48:18.785376Z"Kim, In-Kyun"https://zbmath.org/authors/?q=ai:kim.in-kyun"Won, Joonyeong"https://zbmath.org/authors/?q=ai:won.joonyeongSummary: We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly \(K\)-stable. As a result, they admit an orbifold Kähler-Einstein metric.On the cuspidal locus in the dual varieties of Segre quartic surfaceshttps://zbmath.org/1521.140952023-11-13T18:48:18.785376Z"Honda, Nobuhiro"https://zbmath.org/authors/?q=ai:honda.nobuhiro"Minagawa, Ayato"https://zbmath.org/authors/?q=ai:minagawa.ayatoSummary: Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.Band width and the Rosenberg indexhttps://zbmath.org/1521.190022023-11-13T18:48:18.785376Z"Kubota, Yosuke"https://zbmath.org/authors/?q=ai:kubota.yosukeLet \(M\) be a closed spin manifold. It is necessary to assume the injectivity of the Baum-Connes map in order to get the strongest results relating the nonexistence, stably, of a metric of positive scalar curvature on \(M\) to the higher index theory of \(M\)'s Dirac operator. The paper under review establishes a relationship between a concept closely related to positive scalar curvature and the Dirac operator without assuming the injectivity of the Baum-Connes map. One version of the relationship proven in this paper is: if \(M\) has infinite \(\mathcal K \mathcal O\)-width, the Rosenberg index of \(M\) is not zero. \(\mathcal K \mathcal O\)-width refers to the supremum over possible metrics on \(M\) of the width of bands in \(M\) that have inward boundaries on which the Dirac operators have non-vanishing higher indices. Bands are codimension-zero submanifolds with inward-facing and outward-facing boundaries. Estimates of [\textit{M. Gromov}, in: Foundations of mathematics and physics one century after Hilbert. New perspectives. Cham: Springer. 135--158 (2018; Zbl 1432.53052); Geom. Funct. Anal. 28, No. 3, 645--726 (2018; Zbl 1396.53068); in: Perspectives in scalar curvature. In 2 volumes. Singapore: World Scientific. 1--514 (2023; Zbl 07733259)], adapted to higher-index formulations by \textit{S. Cecchini} [Geom. Funct. Anal. 30, No. 5, 1183--1223 (2020; Zbl 1455.58008)], \textit{R. Zeidler} [J. Differ. Geom. 122, No. 1, 155--183 (2022; Zbl 1515.53053); SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 127, 15 p. (2020; Zbl 1455.19005)], show that infinite \(\mathcal K \mathcal O\)-width prevents \(M\) from admitting a metric of positive scalar curvature.
The author proves the result by showing that the \(K\)-theory class of the maximal equivariant coarse index of the Dirac operator on \(M\)'s universal cover maps by a homomorphism to the corresponding \(K\)-theory class on the universal cover of a band's inward boundary. By definition of the class of bands considered, the latter class is nonzero. The author's methods extend to certain noncompact manifolds and to a multiwidth generalization of band width.
Reviewer: Peter Haskell (Blacksburg)On the capacity dimension of the boundary of \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1521.200912023-11-13T18:48:18.785376Z"Wang, Dawei"https://zbmath.org/authors/?q=ai:wang.daweiThis paper studies the capacity dimension, denoted \(\operatorname{cdim}\), and its relation to the asymptotic dimension, denoted \(\operatorname{asdim}\), of non-spherical buildings and \(\mathrm{CAT(0)}\) groups.
The motivation for this work lies in the fact that, for hyperbolic spaces and groups, the work in [\textit{S. Buyalo}, St. Petersbg. Math. J. 17, No. 2, 267--283 (2006; Zbl 1100.31006); translation from Algebra Anal. 17, No. 2, 70--95 (2005)] established the following bound on the asymptotic dimension
\[
\operatorname{asdim}(X) \leq \operatorname{cdim}(\partial X) + 1.\tag{1}
\]
Meanwhile, \textit{G. Yu} showed in [Ann. Math. (2) 147, No. 2, 325--355 (1998; Zbl 0911.19001)] that any group with finite asymptotic dimension satisfies the Novikov conjecture. This paper explores whether this can be used as a strategy to prove the Novikov conjecture for \(\mathrm{CAT(0)}\) groups, for which the conjecture is open in general. This approach begins with a result by \textit{M. A. Moran} [Geom. Dedicata 183, 123--142 (2016; Zbl 1398.20054)] that states that the boundary of any \(\mathrm{CAT(0)}\) space has finite capacity dimension with respect to conical metrics, which is a class of natural metrics to consider on boundaries of \(\mathrm{CAT(0)}\) spaces. Thus, it would suffice to establish an inequality like inequality (1) for all proper, cocompact \(\mathrm{CAT(0)}\) spaces in order to conclude the Novikov conjecture for all \(\mathrm{CAT(0)}\) groups. The focus of this paper is on establishing good properties of the capacity dimension with respect to conical metrics and confirming this strategy in the well-understood case of buildings.
Theorem 1.3 in this paper states that, if \(X\) is cobounded, hyperbolic, and \(\mathrm{CAT(0)}\), then the capacity dimensions of its boundary with respect to the visual metric, coming from hyperbolic geometry, and the conical metric coming from \(\mathrm{CAT(0)}\) geometry coincide.
Theorem 1.4 specializes to the case of buildings and shows that, if \(\Delta\) is a non-spherical building and \(\Sigma\) an apartment of \(\Delta\), then \(\operatorname{cdim}(\partial \Delta) = \operatorname{cdim}(\partial \Sigma)\). Using results from [\textit{J. Dymara} and \textit{T. Schick}, Russ. J. Math. Phys. 16, No. 3, 409--412 (2009; Zbl 1177.20049)] together with the fact that the relation between asymptotic and capacity dimension is well-understood for hyperbolic and Euclidean space, the author concludes that inequality (1) holds for non-spherical buildings.
The final section of the paper addresses the question how one may attempt to establish inequality (1) for more general \(\mathrm{CAT(0)}\) spaces. The strategy is to use a Hurewicz-type mapping theorem for a map whose preimages are approximations of the boundary in the \(\mathrm{CAT(0)}\) space. The key obstacle seems to be that one has to control the divergence of geodesic rays, which requires one to juggle between arguments that apply for curvature close to zero and arguments that apply when the curvature tends to \(-\infty\). An interesting intermediate question might be whether the arguments in Section 5 can be made precise if one additionally assumes a lower curvature bound in the sense of Alexandrov.
With all the necessary terminology introduced in Section 2 (with a minor error in Definition 2.8: \(\mathcal{L}(\mathcal{U}, x)\) should be the minimum of \(\sup \{ d(x, X \setminus U) \mid U \in \mathcal{U} \}\) and \(\sup_{U \in \mathcal{U}} \{ \operatorname{diam}(U) \mid x \in U \}\)) and an outline of the obstacles that need to be dealt with in Section 5, this paper provides a useful starting point for a potential proof of the Novikov conjecture for \(\mathrm{CAT(0)}\) groups.
Reviewer: Merlin Incerti-Medici (Wien)The isometry groups of Lorentzian three-dimensional unimodular simply connected Lie groupshttps://zbmath.org/1521.220052023-11-13T18:48:18.785376Z"Boucetta, Mohamed"https://zbmath.org/authors/?q=ai:boucetta.mohamed"Chakkar, Abdelmounaim"https://zbmath.org/authors/?q=ai:chakkar.abdelmounaimIn this paper, the authors determine the full isometry groups of all three-dimensional unimodular Lie groups endowed with a left invariant Lorentzian metric. One of the consequences of their result is that, in some cases, the isometry group is bigger than the subgroup of automorphisms that preserve the metric.
Reviewer: Mohammed Guediri (Riyadh)A Cartan decomposition for Gelfand pairs and induction of spherical functionshttps://zbmath.org/1521.220152023-11-13T18:48:18.785376Z"Tanaka, Yuichiro"https://zbmath.org/authors/?q=ai:tanaka.yuichiroSummary: In this article we show a Cartan decomposition for reductive Riemannian Gelfand pairs and an induction of spherical functions for Riemannian Gelfand pairs. With the induction we find that the property of the symmetry of spherical functions, which is known for Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension. A Fourier transform of a positive function for a Riemannian Gelfand pair with abelian unipotent radical is also given under some condition on its support by using the symmetry of spherical function.Homogenization of random quasiconformal mappings and random Delauney triangulationshttps://zbmath.org/1521.300292023-11-13T18:48:18.785376Z"Ivrii, Oleg"https://zbmath.org/authors/?q=ai:ivrii.oleg-v"Marković, Vladimir"https://zbmath.org/authors/?q=ai:markovic.vladimir|markovic.vladimir-mLet \(\lambda\) be a probability measure on the standard unit disk of the complex plane \({\mathbb{C}}\). The authors randomly assign a complex number in the unit disk for each cell in a square grid in the complex plane according to the measure \(\lambda\). The collection of these numbers defines a Beltrami coefficient \(\mu(z)\) on \({\mathbb{C}}\) which is constant on the cells of the grid. The Beltrami equation \(\bar\partial w(z)=\mu(z)\partial w(z)\) has a unique injective solution \(w^\mu\) that fixes \(0\), \(1\) and \(\infty\). The authors call \(w^\mu\) a random quasiconformal mapping, but point out that \(w^\mu\) may not be quasiconformal. The first main result of the paper says that if the mesh size of the grid is small, then with high probability, \(w^{\mu}\) is close to an affine transformation \(A_\lambda\) determined by the measure \(\lambda\).
A circle packing is a collection of circles in \({\mathbb{C}}\) with disjoint interiors. By the Koebe-Andreev-Thurston circle packing theorem, any finite triangulation of a topological disk admits a maximal circle packing whose boundary circles are horocycles. A discrete set \(V\) of points in \({\mathbb{C}}\) determines a Voronoi tessellation. This means that \({\mathbb{C}}\) can be written as a union of sets \(F_x\), \(x\in V\), where \(F_x\) consists of all points \(z\in {\mathbb{C}}\) for which \(\min_{y\in V}\vert y-z\vert=\vert x-z\vert\). If the points in \(V\) are in general position, then the Delauney triangulation is the dual graph to the Voronoi tessellation. The union of all the triangles in the Delauney triangulation is the convex hull of \(V\), and hence a topological disk.
Let \(\Omega\subset {\mathbb{C}}\) be a simply connected domain bounded by a \(\mathrm{C}^1\)-curve. The authors randomly choose \(N\geq 1\) points in \(\Omega\) with respect to a Lebesgue measure. For technical reasons, they also choose \(\asymp \sqrt{N}\) equally spaced points on the boundary \(\partial\Omega\).They define the random Delauney triangulation as the union of the Delauney triangles contained in \(\Omega\). Let \(\varphi_{\mathcal{P}}\) denote the circle packing map of the suitably normalized maximal circle packing of the random Delauney triangulation. Kenneth Stephenson has suggested that when \(N\) is large, then with high probability, \(\varphi_{\mathcal{P}}\) approximates a conformal map \(\varphi\colon \Omega \to {\mathbb{D}}\). The second main result of the paper is a proof of this conjecture.
Reviewer: Marja Kankaanrinta (Helsinki)The Weil-Petersson current on Douady spaceshttps://zbmath.org/1521.320102023-11-13T18:48:18.785376Z"Axelsson, Reynir"https://zbmath.org/authors/?q=ai:axelsson.reynir"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgSummary: The Douady space of compact subvarieties of a Kähler manifold is equipped with the Weil-Petersson current, which is everywhere positive with local continuous potentials, and of class \(C^\infty\) when restricted to the locus of smooth fibers. There a Quillen metric is known to exist, whose Chern form is equal to the Weil-Petersson form. In the algebraic case, we show that the Quillen metric can be extended to the determinant line bundle as a singular hermitian metric. On the other hand the determinant line bundle can be extended in such a way that the Quillen metric yields a singular hermitian metric whose Chern form is equal to the Weil-Petersson current. We show a general theorem comparing holomorphic line bundles equipped with singular hermitian metrics which are isomorphic over the complement of a snc divisor \(B\). They differ by a line bundle arising from the divisor and a flat line bundle. The Chern forms differ by a current of integration with support in \(B\) and a further current related to its normal bundle. The latter current is equal to zero in the case of Douady spaces due to a theorem of Yoshikawa on Quillen metrics for singular families over curves.
{{\copyright} 2021 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH}A generalised volume invariant for Aeppli cohomology classes of Hermitian-symplectic metricshttps://zbmath.org/1521.320182023-11-13T18:48:18.785376Z"Dinew, Sławomir"https://zbmath.org/authors/?q=ai:dinew.slawomir"Popovici, Dan"https://zbmath.org/authors/?q=ai:popovici.dan-emanuel|popovici.danA class of compact complex manifolds is introduced. Let $X$ be an $n$-dimensional compact complex manifold. A Hermitian metric $\omega$ on the manifold $X$ is said to be Hermitian symplectic if $\omega$ is the component of bidegree $(1,1)$ of a real $C^\infty$ $d$-closed 2-form $\widetilde\omega$ on $X$. A manifold $X$ admitting such a metric is called a Hermitian symplectic manifold. Hermitian symplectic manifolds form a natural generalization of compact Kähler manifolds.
For every Hermitian symplectic metric $\omega$ on the manifold $X$, an energy functional acting on the metrics in the Aeppli cohomology class of $\omega$ is introduced. It is proved that its critical points must be Kählerian in the case that the manifold $X$ is 3-dimensional. An interpretation of these critical points as maximizers of the volume of the metric in its Aeppli class is considered.
A numerical invariant for any Aeppli cohomology class of Hermitian symplectic metrics that generalizes the volume of a Kählerian class is defined using this energy functional. Some important results related to this invariant are also presented.
Reviewer: Mihail Banaru (Smolensk)Heat kernels for a class of hybrid evolution equationshttps://zbmath.org/1521.350992023-11-13T18:48:18.785376Z"Garofalo, Nicola"https://zbmath.org/authors/?q=ai:garofalo.nicola"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioSummary: The aim of this paper is to construct (explicit) heat kernels for some \textit{hybrid} evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form \({\mathscr{L}}_1 + {\mathscr{L}}_2 - \partial_t\), but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators \({\mathscr{L}}_1 - \partial_t\) and \({\mathscr{L}}_2 - \partial_t\). Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity.Catching all geodesics of a manifold with moving balls and application to controllability of the wave equationhttps://zbmath.org/1521.370342023-11-13T18:48:18.785376Z"Letrouit, Cyril"https://zbmath.org/authors/?q=ai:letrouit.cyrilSummary: We address the problem of catching all speed-1 geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold \((M, g)\) and small parameters \(\varepsilon>0\) and \(v>0\), is it possible to find \(T > 0\) and an absolutely continuous map \(x:[0,T] \to M, t \mapsto x(t)\) satisfying \(\|\dot{x}\|_\infty \leqslant v\) and such that any geodesic of \((M, g)\) traveled at speed 1 meets the open ball \(B_g (x(t), \varepsilon) \subset M\) within time \(T\)? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold \((M, g)\) satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any \(\varepsilon>0\) and \(v>0\), and we give examples of Riemannian manifolds \((M, g)\) for which (GRC) is satisfied. Then, we build an explicit example of a domain \(X \subset \mathbb{R}^2\) (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if \(\varepsilon\) and \(v\) are small enough, \textit{i.e.}, no sufficiently small ball moving sufficiently slowly can catch all geodesics of \(X\).Quantum Bianchi-VII problem, Mathieu functions and arithmetichttps://zbmath.org/1521.370622023-11-13T18:48:18.785376Z"Veselov, A. P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Ye, Y."https://zbmath.org/authors/?q=ai:ye.yiruThis paper is devoted to the geodesic problem on compact threefolds with Riemannian metric of Bianchi-type. This problem is studied in both classical and quantum cases. The authors show that the problem is integrable; they describe explicitly the eigenfunctions of the corresponding Laplace-Beltrami operators in terms of Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. They employ a number-theoretic approach in order to study the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation when some Bianchi-type metrics are taken into account.
Reviewer: Oleg Karpenkov (Liverpool)On a class of quadratic conservation laws for Newton equations in Euclidean spacehttps://zbmath.org/1521.370752023-11-13T18:48:18.785376Z"Tsiganov, A. V."https://zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovich"Porubov, E. O."https://zbmath.org/authors/?q=ai:porubov.e-oSummary: We discuss quadratic conservation laws for the Newton equations and the corresponding second-order Killing tensors in Euclidean space. In this case, the complete set of integrals of motion consists of polynomials of the second, fourth, sixth, and so on degrees in momenta, which can be constructed using the Lax matrix related to the hierarchy of the multicomponent nonlinear Schrödinger equation.Some fixed-point theorems for a pair of Reich-Suzuki-type nonexpansive mappings in hyperbolic spaceshttps://zbmath.org/1521.470962023-11-13T18:48:18.785376Z"Valappil, Sreya Valiya"https://zbmath.org/authors/?q=ai:valappil.sreya-valiya"Pulickakunnel, Shaini"https://zbmath.org/authors/?q=ai:pulickakunnel.shainiSummary: In this article, we prove some fixed-point results for a pair of Reich-Suzuki-type nonexpansive mappings in uniformly convex \(W\)-hyperbolic spaces. We introduce a new iterative scheme and establish its convergence to the fixed points of a pair of Reich-Suzuki-type nonexpansive mappings. We illustrate our main result with an example, and using Matlab code, it is observed that our iteration converges faster than the iteration defined by \textit{C. Garodia} et al. [Bull. Iran. Math. Soc. 48, No. 4, 1493--1512 (2022; Zbl 1510.47086)] for a pair of Reich-Suzuki-type nonexpansive mappings. An application is given to substantiate our main result.Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifoldshttps://zbmath.org/1521.490352023-11-13T18:48:18.785376Z"Wang, Yu-Zhao"https://zbmath.org/authors/?q=ai:wang.yuzhao"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjie"Zhang, Xinxin"https://zbmath.org/authors/?q=ai:zhang.xinxinSummary: In this paper, we prove the generalized displacement convexity for nonlinear mobility continuity equation with \(p\)-Laplacian on Wasserstein space over Riemannian manifolds under the generalized McCann condition GMC\((m, n)\). Moreover, we obtain some variational formulae along the Langevin deformation of flows on the generalized Wasserstein space, which is the interpolation between the gradient flow and the geodesic flow. We also establish the connection between the displacement convexity of entropy functionals and the concavity of \(p\)-Rényi entropy powers. As an application, we derive the NIW formula which indicates the relationship between the \(p\)-Rényi entropy powers \({\mathcal{N}}_b\), the Fisher information \({\mathcal{I}}_b\) and the \({\mathcal{W}}\)-entropy.Isometric embeddings of bounded metric spaces in the Gromov-Hausdorff classhttps://zbmath.org/1521.510052023-11-13T18:48:18.785376Z"Ivanov, Alexandr O."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-olegovich"Tuzhilin, Alexey A."https://zbmath.org/authors/?q=ai:tuzhilin.alexey-aThe authors show that any bounded metric space can be embedded isometrically in the Gromov-Hausdorff metric class \(\mathcal{GH}\) (the class of isometry classes of all metric spaces, endowed with the Gromov-Hausdorff distance). This follows from the description of the local geometry of \(\mathcal{GH}\) in a sufficiently small neighbourhood of a generic metric space.
Reviewer: Victor V. Pambuccian (Glendale)Pedal and contrapedal curves of framed immersions in the Euclidean 3-spacehttps://zbmath.org/1521.530032023-11-13T18:48:18.785376Z"Yao, Kaixin"https://zbmath.org/authors/?q=ai:yao.kaixin"Li, Meixuan"https://zbmath.org/authors/?q=ai:li.meixuan"Li, Enze"https://zbmath.org/authors/?q=ai:li.enze"Pei, Donghe"https://zbmath.org/authors/?q=ai:pei.dongheLet \(\gamma: I \rightarrow {\mathbb R}^3\) be a smooth space curve and let \(\Delta := \{(x, y) \in S^2 \times S^2 \; | \; x \cdot y = 0\}\), where \(S^2\) denotes the unit 2-sphere. Then \((\gamma, \nu_1, \nu_2) : I \rightarrow {\mathbb R}^3 \times \Delta\) is called a framed curve, if \(\gamma'(t)\cdot \nu_1 = \gamma'(t)\cdot \nu_2 = 0\) for all \(t \in I\). Moreover, it is called a framed immersion if \((\gamma, \nu_1, \nu_2)\) is an immersion. The frame is completed by the unit tangent vector \({\mu} := \nu_1 \times \nu_2\) which is a multiple of \(\gamma'(t)\). A curve \(\gamma(t)\) is called a frame based curve if there exist unit vectors \(\nu_1\), \(\nu_2\) with \(\nu_1 \cdot \nu_2 = 0\) such that the triple \((\gamma, \nu_1, \nu_2)\) is a framed curve. Regular curves are of course frame based because one can choose the unit principal normal vector \(n_1\) and the unit binormal vector \(n_2\) as the pair \((\nu_1, \nu_2) \in \Delta\). Conversely, if \(\gamma(t)\) is frame based, then the pair \(\nu_1\), \(\nu_2\) can be exchanged by the pair consisting of the principal normal vector \(n_1\) and the unit binormal vector \(n_2\). This means that it suffices to consider frame based curves whose frame is the Frenet frame \((n_1, n_2, {\mu})\).
For a given framed immersion \((\gamma, n_1, n_2)\) and a given point \(p\) in \({\mathbb R}^3\) one has that:
(1) The pedal curve \(\mathcal{P}e_{\gamma, p}\) is the curve whose points are the orthogonal projections of \(p\) on the osculating planes of \(\gamma\). The osculating plane of \(\gamma\) at \(\gamma(t)\) is the plane through \(\gamma(t)\) and parallel to \({\mu}(t)\), \(n_1(t)\);
(2) The contrapedal curve \(\mathcal{CP}e_{\gamma, p}\) is the curve whose points are the orthogonal projections of \(p\) on the normal planes of \(\gamma\). The normal plane of \(\gamma\) at \(\gamma(t)\) is the plane through \(\gamma(t)\) and parallel to \(n_1(t)\), \(n_2(t)\).
The authors discuss various properties of pedal and contrapedal curves. For instance, they investigate under which conditions the pedal curve (contrapedal curve) of a given frame based curve is again frame based. Formulae for the curvature of pedal and contrapedal curves are derived. Moreover, it is shown that the contrapedal curve \(\mathcal{CP}e_{\gamma, p}\) of a curve \(\gamma\) with respect to a point \(p\) is at the same time the pedal curve with respect to \(p\) of the evolute of \(\gamma\). In analogy, the pedal curve \(\mathcal{P}e_{\gamma, p}\) of \(\gamma\) with respect to \(p\) is at the same time the contrapedal curve with respect to \(p\) of any of the involutes of \(\gamma\).
The paper is rounded off with 3 examples.
Reviewer: Anton Gfrerrer (Graz)Geometry of two-dimensional surfaces in space \(^2\mathbb{R}_5 \)https://zbmath.org/1521.530042023-11-13T18:48:18.785376Z"Artykbaev, A."https://zbmath.org/authors/?q=ai:artykbaev.abdullaaziz|artykbaev.a-a"Mamadaliyev, B. M."https://zbmath.org/authors/?q=ai:mamadaliyev.b-mSummary: The article is devoted to study the geometry of a two-dimensional surface in a five-dimensional pseudo-Euclidean space. The study of the geometry of a five-dimensional pseudo-Euclidean space is appealing, because de Sitter space is realized on the sphere of this space. Galileo's geometry appears in subspaces with an isotropic part. For the chosen two-dimensional surface, the first and second quadratic forms of the surfaces are determined. A two-dimensional surface is said to be complete, if it is not immersed in a four-dimensional plane. The existence of a complete spherically univalent two-dimensional surface is proved.Quasiconformal harmonic graphshttps://zbmath.org/1521.530052023-11-13T18:48:18.785376Z"Kalaj, David"https://zbmath.org/authors/?q=ai:kalaj.david"Vujadinović, Djordjije"https://zbmath.org/authors/?q=ai:vujadinovic.djordjijeThe authors mainly consider harmonic quasiconformal surfaces in \(\mathbb{R}^{3}\). They first prove that the harmonic graph \(\Sigma\), whose projection is the unit disk, can have arbitrary small (negative) Gaussian curvature at a point \(a\in\Sigma\). They prove the boundedness property for quasiconformal harmonic graphs. The second result in this paper is an extension of Bernstein's theorem that asserts that an entire graph that allows a harmonic quasiconformal parametrization must be a plane. On the other hand, if \(f:C\rightarrow M\) is a conformal harmonic complete embedded parametrization of a minimal surface, then \(M\) can be either a plane or a helicoid. (This is one of the main theorems in the modern theory of minimal surfaces.) The authors show that the helicoid and planar surfaces are not unique in the class of complete embedded quasiconformal harmonic surfaces as it is the case for minimal surfaces. The last result in this paper is related to quasiconformal harmonic immersions. At the end of the paper the authors formulate some open questions.
Reviewer: Shengjin Huo (Tianjin)Surfaces of constant anisotropic mean curvature with free boundary in revolution surfaceshttps://zbmath.org/1521.530072023-11-13T18:48:18.785376Z"Barbosa, Ezequiel"https://zbmath.org/authors/?q=ai:barbosa.ezequiel-r"Carvalho Silva, Lucas"https://zbmath.org/authors/?q=ai:carvalho-silva.lucasSummary: In this paper we consider immersions with constant anisotropic mean curvature (CAMC) of a smooth oriented connected and compact surface \(\varSigma \), with non-empty boundary \(\partial \varSigma \), in a region \(\varOmega\) such that the boundary \(\partial \varOmega\) is a rotational surface. We prove that, under a suitable condition on the anisotropic function, the flat disks are the only free boundary CAMC immersions in \(\varOmega \). Moreover, we study which disks are stable. Finally, we consider an interesting result that allows us to build a wide variety of examples of Wulff Shape.Submanifolds of cosymplectic statistical-space-formshttps://zbmath.org/1521.530112023-11-13T18:48:18.785376Z"Mandal, Pradip"https://zbmath.org/authors/?q=ai:mandal.pradip"Hui, Shyamal Kumar"https://zbmath.org/authors/?q=ai:hui.shyamal-kumarAuthors' abstract: In this paper, invariant and anti-invariant submanifolds of cosymplectic statistical-space-forms are considered. Among others a generalized Wintgen inequality on Legendrian submanifolds of such space forms is obtained.The characterizations on a class of weakly weighted Einstein-Finsler metricshttps://zbmath.org/1521.530122023-11-13T18:48:18.785376Z"Cheng, Xinyue"https://zbmath.org/authors/?q=ai:cheng.xinyue"Cheng, Hong"https://zbmath.org/authors/?q=ai:cheng.hongThe authors introduce weakly weighted Einstein-Finsler metrics and prove that they must be of isotropic S-curvature with respect to the Busemann-Hausdorff volume form under a certain condition about the weight constants. Further they characterize weakly-weighted Einstein-Kropina metrics via their navigation expression.
Reviewer: V. K. Chaubey (Gorakhpur)On \(L\)-reducible spherically symmetric Finsler metricshttps://zbmath.org/1521.530132023-11-13T18:48:18.785376Z"Tayebi, A."https://zbmath.org/authors/?q=ai:tayebi.ali|tayebi.akbar|tayebi.arash|tayebi.amin|tayebi.abdelhamid"Barati, F."https://zbmath.org/authors/?q=ai:barati.farzan|barati.fahimehSummary: In this paper, we study one of the oldest open problems in Finsler geometry which was introduced by \textit{M. Matsumoto} and \textit{H. Shimada} [Rep. Math. Phys. 12, 77--87 (1977; Zbl 0375.53011)] about the existence of a concrete \(L\)-reducible Finsler metric that is not \(C\)-reducible. To spot such a Finsler metric, we study the class of spherically symmetric Finsler metrics. We prove two rigidity theorems for spherically symmetric Finsler metrics. First, we prove that every spherically symmetric Finsler metric is semi-\(C\)-reducible. Second, we show that every non-Riemannian spherically symmetric Finsler metric is a generalized \(L\)-reducible metric. Finally, under a particular condition, we prove that every non-Riemannian \(L\)-reducible spherically symmetric Finsler metric on a manifold of dimension \(n \geq 3\) must be a Randers metric. This result provides a negative answer to Matsumoto-Shimada's problem in the class of spherically symmetric Finsler metrics.Geometric theory of Weyl structureshttps://zbmath.org/1521.530142023-11-13T18:48:18.785376Z"Čap, Andreas"https://zbmath.org/authors/?q=ai:cap.andreas"Mettler, Thomas"https://zbmath.org/authors/?q=ai:mettler.thomasGiven a parabolic geometry on a smooth manifold \(M\), the authors study a natural affine bundle \(A\to M\) whose smooth sections can be identified with Weyl structures.
In particular, the paper considers torsion-free AHS structures:
Let \(g\) be a semisimple Lie algebra endowed with a \(|1|\)-grading, i.e., with a decomposition
\[ g= g_{-1}\oplus g_0\oplus g_1 \]
into a direct sum of linear subspaces such that:
(1) \([ g_i, g_j]\subset g_{i+j}\) setting \( g_\ell=\{0\}\) if \(|\ell|>1\);
(2) No simple ideal of \( g\) is contained in \( g_0\).
Then \( \widetilde p= g_0+ g_1\) is a Lie subalgebra of \( g\). Let \(G\) be a Lie group with Lie algebra \(g\). The normalizer of \(\widetilde p\) in \(G\) has Lie algebra \({p}\). Choosing a closed subgroup \(P\subset G\) lying between this normalizer and its connected component of the identity, a parabolic geometry of type \((G,P)\) is a Cartan geometry \((p:\mathcal{G}\to M,\omega)\) of type \((G,P)\).
Definition. For the parabolic geometry \((p:\mathcal G\to M,\omega)\) the associated bundle of Weyl structures is
\[ \pi:A:=\mathcal{G}\times_P(P/G_0)\to M, \]
where \(G_0\subset P\) is the closed subgroup of elements whose adjoint action preserves the grading of \(g\).
Weyl structures for parabolic geometries were originally introduced by \textit{A. Čap} and \textit{J. Slovák} [Math. Scand. 93, No. 1, 53--90 (2003; Zbl 1076.53029)] as \(G_0\)-equivariant sections of the natural projection \(q:\mathcal G\to\mathcal G_0:=\mathcal G/P_+\). Such sections correspond to smooth sections of \(\pi:A\to M\) (see \S2.2), such that the space of sections of \(A\to M\) can be naturally identified with the space of Weyl structures for the geometry \((p:\mathcal G\to M,\omega)\).
The initial parabolic geometry is shown to define a reductive Cartan geometry on \(A\), inducing an almost bi-Lagrangian structure \((\Omega,L^+,L^-)\) on \(A\) and a compatible linear connection on \(TA=L^-\oplus L^+\) (see \S2.3 and \S3.1). The skew symmetric bilinear bundle map \(\Omega\) defines an almost symplectic structure. One of the main theorems concerns the case when it is symplectic.
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a parabolic geometry of type \((G,P)\) and \(\pi:A\to M\) its associated bundle of Weyl structures. Then the natural 2-form \(\Omega\in\Omega^2(A)\) is closed if and only if \((G,P)\) corresponds to a \(|1|\)-grading and the Cartan geometry \((p:\mathcal{G}\to M,\omega)\) is torsion-free.
The canonical almost bi-Lagrangian structure on the bundle \(A\to M\) also induces a canonical nondegenerate bilinear form \(h\).
Theorem. For any torsion-free AHS structure, the pseudo-Riemannian metric \(h\) induced by the canonical almost bi-Lagrangian structure on \(A\to M\) is an Einstein metric with nonzero scalar curvature.
A Weyl structure of a torsion-free AHS structure is Lagrangian if \(s:M\to(A,\Omega)\) is a Lagrangian submanifold, and nondegenerate if \(s:M\to(A,h)\) is a nondegenerate submanifold. Lagrangian Weyl structures that lead to totally geodesic submanifolds \(s(M)\subset A\) are characterized as follows (the Rho-tensor is introduced in \S2.2):
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a torsion-free AHS structure and \(\pi:A\to M\) its bundle of Weyl structures. Let \(s:M\to A\) be a smooth section corresponding to a Lagrangian Weyl structure, let \(\nabla^s\) and \(\mathsf{P}^s\) denote the corresponding Weyl connection and Rho-tensor, respectively. Then the following conditions are equivalent:
(1) The submanifold \(s(M)\subset A\) is totally geodesic for the canonical connection \(D\) (\S2.3);
(2) The submanifold \(s(M)\subset A\) is totally geodesic for the Levi-Civita connection of \(h\);
(3) \(\nabla^s\mathsf P^s = 0\).
This provides a connection to Einstein metrics and reductions of projective holonomy (\S3.4).
Moreover, given nondegeneracy, a universal formula for the second fundamental form of this image is obtained. The second fundamental forms of \(s:M\to(A,h)\) with respect to \(D\) and the Levi-Civita connection of \(h\) are, respectively, denoted by \(\mathrm{I\!I}^s_D\) and \(\mathrm{I\!I}^s_h\).
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a torsion-free AHS structure and \(\pi:A\to M\) its bundle of Weyl structures. Let \(s:M\to A\) be a nondegenerate, Lagrangian Weyl structure with Weyl connection \(\nabla^s\) and Rho-tensor \(\mathsf{P}\). Then \[ \mathrm{I\!I}^s_D=-\frac12\mathsf{P}^{ka}\nabla^s_i\mathsf{P}_{ja} \] and \[ \mathrm{I\!I}^s_h = -\frac12\mathsf{P}^{ka}\,(\nabla^s_i\mathsf{P}_{ja}+\nabla^s_j\mathsf{P}_{ia}-\nabla^s_a\mathsf{P}_{ij}). \]
In \S4, relations between the geometry on \(A\) and nonlinear invariant partial differential equations are discussed. For locally flat projective structures, interrelations to solutions of a projectively invariant Monge-Ampère equation are obtained. Theorem 4.4 relates Calabi's equation to an equation for a minimal Lagragian Weyl structure. As a consequence, the following statement is obtained:
Corollary. Let \((M, [\nabla])\) be a closed oriented locally flat projective manifold. Then \([\nabla]\) is properly convex if and only if it arises from a minimal Lagrangian Weyl structure whose Rho-tensor is positive definite.
Reviewer: Andreas Vollmer (Hamburg)Relative connections on principal bundles and relative equivariant structureshttps://zbmath.org/1521.530152023-11-13T18:48:18.785376Z"Poddar, Mainak"https://zbmath.org/authors/?q=ai:poddar.mainak"Singh, Anoop"https://zbmath.org/authors/?q=ai:singh.anoopSummary: We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.Holonomy pseudogroups of manifolds over Weil algebrashttps://zbmath.org/1521.530162023-11-13T18:48:18.785376Z"Shurygin, V. V."https://zbmath.org/authors/?q=ai:shurygin.vadim-v-jun"Zubkova, S. K."https://zbmath.org/authors/?q=ai:zubkova.svetlana-kSummary: The notion of the holonomy pseudogroup on a total immersed transversal is extended to the case of complete foliated smooth manifolds over a Weil algebra \({\mathbf{A}}\) modelled on \({\mathbf{A}} \)-modules of the type \({\mathbf{A}}^n\oplus{\mathbf{B}}^m \), where \({\mathbf{B}}\) is a quotient algebra of \({\mathbf{A}} \). It is proved that the holonomy pseudogroup determines a complete \({\mathbf{A}} \)-smooth manifold up to \({\mathbf{A}} \)-diffeomorphism, and examples of application of holonomy pseudogroups are presented.De Rham cohomology and semi-slant submanifolds in metallic Riemannian manifoldshttps://zbmath.org/1521.530172023-11-13T18:48:18.785376Z"Gök, Mustafa"https://zbmath.org/authors/?q=ai:gok.mustafaSummary: In this paper, we deal with the de Rham cohomology of semi-slant submanifolds in metallic Riemannian manifolds. Some necessary conditions for a semi-slant submanifold of metallic Riemannian manifolds are given to define a well-defined canonical de Rham cohomology class. Also, the non-triviality of such a cohomology class is discussed. Finally, an example is constructed to illustrate the main idea of the paper.On the rigidity of the Sasakian structure and characterization of cosymplectic manifoldshttps://zbmath.org/1521.530182023-11-13T18:48:18.785376Z"Patra, Dhriti Sundar"https://zbmath.org/authors/?q=ai:patra.dhriti-sundar"Rovenski, Vladimir"https://zbmath.org/authors/?q=ai:rovenskii.vladimir-yuzefovichSummary: We introduce new metric structures on a smooth manifold (called ``weak'' structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures \((\varphi, \xi, \eta, g)\) and allow us to take a fresh look at the classical theory and find new applications. This assertion is illustrated by generalizing several well-known results. It is proved that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. It is shown that a weak almost contact structure with parallel tensor \(\varphi\) is a weak cosymplectic structure and an example of such a structure on the product of manifolds is given. Conditions are found under which a vector field is a weak contact vector field.Conformal \(\eta\)-Ricci soliton in Lorentzian para Kenmotsu manifoldshttps://zbmath.org/1521.530192023-11-13T18:48:18.785376Z"Prasad, Rajendra"https://zbmath.org/authors/?q=ai:prasad.rajendra"Kumar, Vinay"https://zbmath.org/authors/?q=ai:kumar.vinay-p|kumar.vinay-b-yThe paper deals with conformal \(\eta \)-Ricci solitons on a Lorentzian para-Kenmotsu manifold \((M, g)\). This notion involves a vector field \(V\), a number \(\lambda \in \mathbb{R}\) and a time-dependent scalar field \(p=p(x, t)\), \(x\in M\). Only two examples are discussed: in the first \(\lambda =\frac{p}{2}-\frac{14}{5}\), while in the second \(\lambda =\frac{p}{2}-\frac{2}{3}\). So, a natural question is about the nature of \(p\), a fact that is not explained by the authors.
Reviewer: Mircea Crâşmăreanu (Iaşi)CR-submanifolds of a golden semi-Riemannian space formhttps://zbmath.org/1521.530202023-11-13T18:48:18.785376Z"Verma, Sapna"https://zbmath.org/authors/?q=ai:verma.sapna"Ahmad, Mobin"https://zbmath.org/authors/?q=ai:ahmad.mobinThe paper studies CR-submanifolds in a golden semi-Riemannian space form \(\bar{M}\). Sectional and holomorphic sectional curvatures of \(M\) are computed and a necessary condition for \(M\) to be \(D^{\perp }\)-totally geodesic is obtained. An example is provided for \(\bar{M}=\mathbb{R}^{10}_5\).
Reviewer: Mircea Crâşmăreanu (Iaşi)Asymptotic behaviour of the sphere and front of a flat sub-Riemannian structure on the Martinet distributionhttps://zbmath.org/1521.530212023-11-13T18:48:18.785376Z"Bogaevsky, Ilya A."https://zbmath.org/authors/?q=ai:bogaevskij.ilya-aleksandrovichSummary: The sphere and front of a flat sub-Riemannian structure on the Martinet distribution are surfaces with nonisolated singularities in three-dimensional space. The sphere is a subset of the front; it is not subanalytic at two antipodal points (the poles). The asymptotic behaviour of the sub-Riemannian sphere and Martinet front are calculated at these points: each surface is approximated by a pair of quasihomogeneous surfaces with distinct sets of weights in a neighbourhood of a pole.Variational problems concerning sub-Finsler metrics in Carnot groupshttps://zbmath.org/1521.530222023-11-13T18:48:18.785376Z"Essebei, Fares"https://zbmath.org/authors/?q=ai:essebei.fares"Pasqualetto, Enrico"https://zbmath.org/authors/?q=ai:pasqualetto.enricoLet \({G}\) be a Carnot group equipped with a left-invariant sub-Riemannian metric \(\langle \cdot , \cdot \rangle\) on the horizontal tangent bundle \(H {G}\) induced by the first stratum of its Lie algebra, and denote by \(d_{cc}\) the induced Carnot-Carathéodory distance on \({G}\). Let \(\Omega \subset {G}\) be open. The main objects of interest in this paper are distance functions \(d\) on \(\Omega\) which are geodesic and bi-Lipschitz equivalent to \(d_{cc}\).
Section 4 is concerned with convergence properties for such distances. Given a distance \(d\) on \(\Omega\) satisfying the above conditions, consider the following functionals:
(1) For a Lipschitz path \(\gamma \in \mathrm{Lip}([0,1], \Omega)\), let \(L_d(\gamma)\) be the length of \(\gamma\) with respect to \(d\);
(2) For a positive Borel measure \(\mu\) on \(\Omega \times \Omega\), let \(J_d(\mu) = \int d(x,y) \mu(dx, dy)\). When \(\mu\) is a probability measure, \(J_d(\mu)\) can be interpreted as a coupling of two probability measures on \(\Omega\), and then it represents the transportation cost of this coupling.
Now let \(d_n\) be a sequence of such distances on \(\Omega\), with \(d\) another such distance, and suppose that their bi-Lipschitz constants are uniformly bounded by some \(\alpha \ge 1\). (In the paper, the latter condition is easy to overlook, as it is implicit in the definition of the class \(\mathcal{D}_{cc}(\Omega)\).) Define the functionals \(L_n, J_n\) accordingly from \(d_n\). In Theorem 4.4, the authors show that the following conditions are equivalent:
(1) \(d_n \to d\) uniformly on compact subsets of \(\Omega \times \Omega\);
(2) The sequence \(L_n\) is \(\Gamma\)-convergent to \(L\). This means that if \(\gamma_n \to \gamma\) uniformly on \([0,1]\), then \(L(\gamma) \le \liminf L_n(\gamma_n)\); and for each path \(\gamma\) there exists a sequence \(\gamma_n\) with \(\gamma_n \to \gamma\) uniformly and \(\limsup L_n(\gamma_n) \le L(\gamma)\);
(3) The sequence \(J_n\) is \(\Gamma\)-convergent to \(J\) in a similar sense. Here we ask for the sequences \(\mu_n\) to converge to \(\mu\) in the weak-* topology.
Moreover, if \(\Omega\) is bounded, we can strengthen (3) to \(J_n(\mu_n) \to J(\mu)\) whenever \(\mu_n \to \mu\) in the weak-* topology.
The other main topic of the paper is the relationship between geodesic distance functions on \(\Omega\) and sub-Finsler convex metrics. Here, a sub-Finsler convex metric on \(\Omega\) is a measurable function \(\varphi : H \Omega \to [0, \infty)\) on the horizontal distribution, which is a norm on each fiber \(H_x\). As with the distance functions, the authors restrict attention to those sub-Finsler convex metrics which are bi-Lipschitz equivalent to the sub-Riemannian metric, so that for some \(\alpha \ge 1\) we have \(\frac{1}{\alpha} \|v\| \le \varphi(v) \le \alpha \|v\|\), where \(\|v\|^2 = \langle v, v \rangle\). Such a \(\varphi\) induces a dual norm \(\varphi^\star\) on the dual bundle \(H^* \Omega\), which may be identified again with \(H \Omega\) via the sub-Riemannian metric.
A distance function \(d\) may be differentiated along horizontal directions to produce a sub-Finsler convex metric \(\varphi_d\) (Theorem 3.9). Conversely, given a sub-Finsler convex metric \(\varphi\), there are two natural ways to produce a distance function. On the one hand, we have the intrinsic distance defined via lengths of paths, as
\[
d_\varphi(x,y) = \inf_\gamma \int_0^1 \varphi(\dot{\gamma}(t))\,dt
\]
with the infimum taken as usual over horizontal paths joining \(x\) and \(y\). On the other hand, we can define
\[
\delta_\varphi(x,y) = \sup\{ |f(x) - f(y)| : \varphi(\nabla f) \le 1\}
\]
where \(\nabla\) denotes the horizontal gradient with respect to the reference sub-Riemannian metric \(\langle \cdot, \cdot \rangle\). This construction is in some sense dual to the intrinsic distance, as will be seen.
The results of the paper include:
(1) If \(d\) is a geodesic distance that is bi-Lipschitz equivalent to \(d_{cc}\), it is the intrinsic distance of its metric derivative: \(d = d_{\varphi_d}\) (Theorem 3.5);
(2) If we begin with a sub-Finsler convex metric \(\psi\) and take the metric derivative \(\varphi_{d_\psi}\) of its intrinsic distance, we do not in general recover \(\psi\). However, we do have \(\varphi_{d_\psi} \le \psi\) on almost every fiber \(H_x\), and on every fiber if \(\psi\) is upper semicontinuous. We obtain the opposite inequality if \(\psi\) is lower semicontinuous (Theorem 5.9);
(3) For any sub-Finsler convex metric \(\varphi\), we have \(\delta_\varphi \le d_{\varphi^\star}\), with equality if \(\varphi\) is lower semicontinuous or upper semicontinuous (Theorems 5.11 and 5.12);
(4) If \(\varphi\) is a sub-Finsler convex metric which is upper semicontinuous, then \(\varphi(\nabla f)\) equals the local Lipschitz constant of \(f\) with respect to \(\delta_\varphi\) almost everywhere.
Reviewer: Nathaniel Eldredge (Greeley)Blowups and blowdowns of geodesics in Carnot groupshttps://zbmath.org/1521.530232023-11-13T18:48:18.785376Z"Hakavuori, Eero"https://zbmath.org/authors/?q=ai:hakavuori.eero"Le Donne, Enrico"https://zbmath.org/authors/?q=ai:le-donne.enricoLet \(G\) be a Carnot group of step \(s\) equipped with a left-invariant sub-Finsler metric. Given a Lipschitz curve \(\gamma\) in \(G\), one can construct its blow-up in a natural way: Supposing for instance that \(\gamma(0) = 1_G\) is the identity element, we let \(\gamma_h(t) = \delta_{1/h}(\gamma(ht))\), so that as \(h \to 0\), we are dilating the image of the curve while slowing down time to keep the speed approximately constant. Any limit of a sequence \(\gamma_{h_n}\) as \(h_n \to 0\), in the topology of uniform convergence on compact subsets of time, is called a \textit{tangent} of \(\gamma\). Every Lipschitz curve has at least one tangent (by Arzelá-Ascoli theorem), but in general it is not unique. We also define asymptotic cones (blow-downs) of \(\gamma\) in a similar way by letting \(h_n \to \infty\) instead.
The first main result of this paper concerns projections of blow-ups. Let \(V_s\) be the top layer of the stratification of the Lie algebra of \(G\), which generates the subgroup \(\exp(V_s) \subset G\). Write \(\pi_{s-1}\) for the quotient map from \(G\) to \(G/\exp(V_s)\). Theorem 1.3 states that if \(\gamma\) is a geodesic and \(\sigma\) is a tangent to \(\gamma\) (which implies that \(\sigma\) is also a geodesic), then \(\pi_{s-1} \circ \sigma\) is a geodesic in \(G / \exp(V_s)\) with respect to the canonical sub-Finsler metric induced from \(G\). Keep in mind that the projection of a geodesic is in general not a geodesic. Iterating this \(s-1\) times, one has the corollary that any \((s-1)\)-fold iterated tangent \(\sigma\) of \(\gamma\) projects to a geodesic in the abelianization \(G / [G,G]\). When our sub-Finsler metric is actually sub-Riemannian, this means that \(\sigma\) is a line (Corollary 1.4).
As an extension of this theorem, by considering nilpotent approximations, the authors show that in a general sub-Riemannian manifold of step \(s\) (not necessarily a Carnot group), any \((s-1)\)-fold iterated tangent of a geodesic is a line (Theorem 1.1).
The last main result, back in the setting of a sub-Finsler Carnot group \(G\), deals with the projection \(\pi \circ \gamma\) into the abelianization \(G / [G,G]\) of an infinite geodesic \(\gamma : \mathbb{R} \to G\) (which is required to be length-minimizing along its entire length). It is shown that if \(\pi \circ \gamma\) is not itself a geodesic, then it still must lie within some distance \(R\) of some hyperplane \(W\) (Theorem 1.5). In the special case where the sub-Finsler metric is actually sub-Riemannian, this implies that all asymptotic cones of \(\gamma\) lie in a proper Carnot subgroup of \(G\) (Corollary 1.6).
Reviewer: Nathaniel Eldredge (Greeley)Nowhere differentiable intrinsic Lipschitz graphshttps://zbmath.org/1521.530242023-11-13T18:48:18.785376Z"Julia, Antoine"https://zbmath.org/authors/?q=ai:julia.antoine"Nicolussi Golo, Sebastiano"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastiano"Vittone, Davide"https://zbmath.org/authors/?q=ai:vittone.davideThe paper gives examples of Lipschitz graphs that are nowhere intrinsically differentiable.
The main theorem reads:
Theorem. Let \( G\) be a Carnot group with stratification \(\bigoplus_{j=1}^{s} V_j\). Let \({W}{ V}\) be a splitting of \(G\) such that \({W}V_2\not\subset [{ W},{W}]\) and there exists \(v_0 \in {V}\cap V_1\) such that \(v_0\not= 0\) and \([v_0,{W}]=0\).
Then there is an intrinsic Lipschitz function \(\phi: {W} \rightarrow { V}\) that is nowhere intrinsically differentiable.
Moreover, \(\phi\) can be constructed in such a way that, for every \(p\in \Gamma_{\phi}\), the following properties hold:
(a) There exist infinitely many different blow-ups of \(\Gamma_{\phi}\) at \(p\);
(b) No blow-up of \(\Gamma_{\phi}\) at \(p\) is a homogeneous subgroup.
Reviewer: Sergei V. Rogosin (Minsk)On the stability of minimal submanifolds in conformal sphereshttps://zbmath.org/1521.530252023-11-13T18:48:18.785376Z"Franz, Giada"https://zbmath.org/authors/?q=ai:franz.giada"Trinca, Federico"https://zbmath.org/authors/?q=ai:trinca.federicoSummary: Given an \(n\)-dimensional Riemannian sphere conformal to the round one and \(\delta \)-pinched, we show that it does not contain any closed stable minimal submanifold of dimension \(2\le k\le n-\delta^{-1}\).Classifying sufficiently connected PSC manifolds in 4 and 5 dimensionshttps://zbmath.org/1521.530262023-11-13T18:48:18.785376Z"Chodosh, Otis"https://zbmath.org/authors/?q=ai:chodosh.otis"Li, Chao"https://zbmath.org/authors/?q=ai:li.chao.5"Liokumovich, Yevgeny"https://zbmath.org/authors/?q=ai:liokumovich.yevgenyThe authors show that if \(M^n\) is a closed smooth manifold of dimension \(4\) or \(5\) admitting a metric of positive scalar curvature and satisfying some topological conditions on the homotopy groups, then a finite cover of \(M\) is homotopy equivalent to \(S^n\) or to a connected sum of \( S^1\times S^n\)'s. They also prove a more general result by assuming the existence of a non-zero degree map \(f:M\to X\) where \(M\) is a closed manifold with metric of positive scalar curvature and \(X\) is of dimension \(4\) or \(5\) satisfying topological conditions.
Reviewer: Georges Habib (Fanar)A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundaryhttps://zbmath.org/1521.530272023-11-13T18:48:18.785376Z"Sicca, Vladmir"https://zbmath.org/authors/?q=ai:sicca.vladmir"Tsogtgerel, Gantumur"https://zbmath.org/authors/?q=ai:tsogtgerel.gantumurSummary: We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.Some characterizations of spheres by conformal vector fieldshttps://zbmath.org/1521.530282023-11-13T18:48:18.785376Z"Ye, Jian"https://zbmath.org/authors/?q=ai:ye.jianSummary: In this paper, we consider conformal characterizations of standard sphere in terms of conformal vector fields on closed Riemannian manifolds. We firstly prove that each closed Riemannian manifold with Ricci curvature being non-negative in certain direction and constant scalar curvature is isometric to standard sphere if and only if it admits a non-trivial closed conformal vector field. In the case of non-constant scalar curvature, we show that each closed Riemannian manifold of dimension two with positive Gauss curvature carrying a non-trivial closed conformal vector field is conformal to a round sphere and we generalize the result to high dimensions in two directions.Poincaré type inequality for hypersurfaces and rigidity resultshttps://zbmath.org/1521.530292023-11-13T18:48:18.785376Z"Alencar, Hilário"https://zbmath.org/authors/?q=ai:alencar.hilario"Batista, Márcio"https://zbmath.org/authors/?q=ai:batista.marcio"Silva Neto, Gregório"https://zbmath.org/authors/?q=ai:silva-neto.gregorioThe authors prove a general Poincaré type inequality on Riemannian manifolds whose sectional curvature is suitably bounded. They then apply this inequality, combined with some other mild assumptions, to prove the following:
(1) Some isoperimetric inequalities for domains of hypersurfaces;
(2) Minimal hypersurfaces of space forms (satisfying suitable decay properties) are foliated by totally geodesic submanifolds;
(3) Hypersurfaces with a determined constant scalar curvature in Einstein manifolds are totally geodesic;
(4) The hyperplane is rigid, in the sense that is the only homothetic self-similar solution to a class of fully nonlinear curvature flows.
Reviewer: Giorgio Saracco (Firenze)Conformal Dirac-Einstein equations on manifolds with boundaryhttps://zbmath.org/1521.530302023-11-13T18:48:18.785376Z"Borrelli, William"https://zbmath.org/authors/?q=ai:borrelli.william"Maalaoui, Ali"https://zbmath.org/authors/?q=ai:maalaoui.ali"Martino, Vittorio"https://zbmath.org/authors/?q=ai:martino.vittorioLet \(M\) be a compact oriented 3-dimensional manifold with nonempty boundary \(\partial M\). Let the spin structure be fixed on \(M\) and \(\partial M\) carry the one canonically induced by the outward normal along \(\partial M\), whatever the Riemannian metric is. A new functional of Gibbons-Hawking-York type is introduced and its critical points within a conformal class are investigated. This functional \(\mathcal{E}\) is defined by
\[
\mathcal{E}(g,\psi):=\int_M R_g\,dv_g+\frac{1}{2}\int_{\partial M}h_g\,d\sigma_g+\int_M\left(\langle D_g\psi,\psi\rangle-\langle\psi,\psi\rangle\right) \,dv_g,
\]
where \(g\) is a Riemannian metric on \(M\) with scalar curvature \(R_g\) and boundary mean curvature \(h_g\), and \(\psi\) is a spinor field on \((M,g)\). As usual, \(D_g\) denotes the Dirac operator associated to the metric \(g\) and the spin structure on \(M\).
The functional \(\mathcal{E}\) extends both the Gibbons-Hawking-York functional, which generalizes the Einstein-Hilbert functional on manifolds with boundary, and the Einstein-Dirac functional on closed \(3\)-manifolds.
The functional \(\mathcal{E}\) is restricted to a conformal class with fixed boundary volume, in which case the metric \(g\) can be expressed in terms of a real function \(u\) and a supplementary summand involving a real nonnegative parameter \(b\). The critical points of that new functional \((u,\psi)\mapsto E^b(u,\psi)\) are first characterized assuming chiral bag boundary conditions for the Dirac operator. The Euler-Lagrange equations involve a linearising operator \(L_g\) which, under the above boundary conditions, turns out to be elliptic and self-adjoint and thus to have a discrete spectrum consisting of real eigenvalues of finite multiplicities. Assuming the Yamabe invariant of \((M,\partial M)\) w.r.t. the chosen conformal class to be positive, which is equivalent to the first eigenvalue of \(L_g\) being positive, the authors consider sequences converging to critical points of \(E^b\) and the corresponding expected blow-up phenomena. The first main result (Theorem 1.1) gives an asymptotic expansion for a Palais-Smale sequence in both cases where the blow-up occurs in the interior and on the boundary of \(M\). Along the way, a rigidity result is established in the case where \(b=0\) and \(u\geq0\) (Theorem 1.4). The second main result (Theorem 1.6) states an Aubin-type inequality for the Yamabe invariant of \((M,\partial M)\) in the case \(b=0\).
The article is structured as follows. After the presentation of the main results in Section 1, preliminaries about the operators and their conformal covariance are discussed in Section 2. A regularity result (Theorem 3.1) is then established in Section 3. Theorem 1.1 is proved in Section 4, while the rigidity result for \(b=0\) is dealt with in Section 5. Section 6 contains the proof of the Aubin-type inequality.
Reviewer: Nicolas Ginoux (Metz)Some characterizations of Bach solitons via Ricci curvaturehttps://zbmath.org/1521.530312023-11-13T18:48:18.785376Z"Cunha, Antonio W."https://zbmath.org/authors/?q=ai:cunha.antonio-wilson"de Lima, Eudes L."https://zbmath.org/authors/?q=ai:de-lima.eudes-leite"Mi, Rong"https://zbmath.org/authors/?q=ai:mi.rongSummary: In this short note we provide some results for Bach solitons under different assumptions. In fact, under either non-negative or non-positive Ricci curvature condition we are able to show that a Bach soliton must be Bach-flat, since it satisfies a finite weighted Dirichlet integral condition or a parabolicity condition jointly with some regularity conditions \(L^\infty\) or \(L^p\) on gradient of the potential function.The weighted Yamabe flow with boundaryhttps://zbmath.org/1521.530322023-11-13T18:48:18.785376Z"Ho, Pak Tung"https://zbmath.org/authors/?q=ai:ho.pak-tung"Shin, Jinwoo"https://zbmath.org/authors/?q=ai:shin.jinwoo"Yan, Zetian"https://zbmath.org/authors/?q=ai:yan.zetianSummary: We introduce a Yamabe-type flow
\[
\begin{cases}
\frac{\partial g}{\partial t} \ = (r^m_{\phi}-R^m_{\phi})g \\
\frac{\partial \phi}{\partial t} \ = \frac{m}{2}(R^m_{\phi}-r^m_{\phi})
\end{cases}
\text{ in }M \text{ and } H^m_{\phi} = 0 \quad\text{on }\partial M
\]
on a smooth metric measure space with boundary \((M,g, v^mdV_g,v^mdA_g,m) \), where \(R^m_{\phi}\) is the associated weighted scalar curvature, \( r^m_{\phi}\) is the average of the weighted scalar curvature, and \(H^m_{\phi}\) is the weighted mean curvature. We prove the long-time existence and convergence of this flow.Generalized surgery on Riemannian manifolds of positive Ricci curvaturehttps://zbmath.org/1521.530332023-11-13T18:48:18.785376Z"Reiser, Philipp"https://zbmath.org/authors/?q=ai:reiser.philippThe paper is concerned with constructing metrics of positive Ricci curvature on simply connected manifolds. By the seminal work of \textit{J.-P. Sha} and \textit{D. Yang} [Proc. Symp. Pure Math. 54, 529--538 (1993; Zbl 0788.53029)] and \textit{D. Wraith} [J. Differ. Geom. 45, No. 3, 638--649 (1997; Zbl 0910.53027); J. Reine Angew. Math. 501, 99--113 (1998; Zbl 0915.53018)], the property of admitting positive Ricci curvature is preserved under certain surgeries which implies that the boundaries of plumbings of disc bundles over spheres support a metric of positive Ricci curvature, provided that base and fiber of these bundles have dimension at least 3.
In the present article, this result is generalized for disc bundles over manifolds admitting \textit{core metrics}. More precisely, a plumbing of linear sphere bundles over manifolds admitting core-metrics admits a metric of positive Ricci curvature. Many manifolds admitting core-metrics are known by work of \textit{B. L. Burdick} [Differ. Geom. Appl. 62, 212--233 (2019; Zbl 1417.53037)], for example all simply connected spheres and complex, quaternionic and octonionic projective spaces fall into this class. In this paper, more examples of manifolds admitting such metrics are constructed.
As an application, the author constructs a new infinite family of simply connected \(6\)-manifolds admitting positive Ricci curvature metrics.
Reviewer: Georg Frenck (Augsburg)Ambient prime geodesic theorems on hyperbolic 3-manifoldshttps://zbmath.org/1521.530342023-11-13T18:48:18.785376Z"Dever, Lindsay"https://zbmath.org/authors/?q=ai:dever.lindsay"Milićević, Djordje"https://zbmath.org/authors/?q=ai:milicevic.djordjeThe study of the length spectrum of geodesics on a hyperbolic manifold is an interesting and important problem, which has been the subject of a number of papers. Asymptotic estimates for the growth of the number of nonequivalent classes of primitive geodesics of length tending to infinity on such manifolds are known. In the present article, the authors consider a natural complex generalization of the length of a geodesic: the real part is the usual length, and the imaginary part is the angle around the geodesic, which corresponds to parallel translation along it (the holonomy of the geodesic). The authors consider the cutoff of the set of classes of primitive geodesics in both parameters, the length and angle of the holonomy. An asymptotic estimate is given for the number of classes of primitive geodesics on a three-dimensional hyperbolic manifold obtained by a given truncation in the two parameters.
Reviewer: Anton Galaev (Hradec Králové)Graph comparison meets Alexandrovhttps://zbmath.org/1521.530352023-11-13T18:48:18.785376Z"Lebedeva, N. D."https://zbmath.org/authors/?q=ai:lebedeva.n-d"Petrunin, A. M."https://zbmath.org/authors/?q=ai:petrunin.antonGraph comparison is a certain type of condition on a metric space encoded by a finite graph. Suppose that \(\Gamma\) is a graph with vertices \(v_1,\ldots ,v_n\). The authors write \(v_i \sim v_j\) (or \(v_i\not\sim v_j\)) if \(v_i\) is adjacent (or nonadjacent) to \(v_j\).
A metric space \(X\) meets the \(\Gamma\)-comparison if for every \(n\) points in \(X\) labeled by vertices of \(\Gamma\) there is a model configuration \(\tilde{v}_1, \ldots, \tilde{v}_n\) in some Hilbert space \(\mathbb{H}\) such that:
\begin{align*}
v_i \sim v_j&\implies | v_i \sim v_j|_{\mathbb{H}}\leq | v_i-v_j|_{X},\\
v_i \not\sim v_j&\implies | v_i \sim v_j|_{\mathbb{H}}\geq | v_i-v_j|_{X},
\end{align*}
where \(| \cdot -\cdot |_X\) is the metric in \(X\). By \(T_3\) the authors denote \(K_{1,3}\) and by \(C_4\) a four-cycle. The following theorem is proved:
Theorem. Let \(\Gamma\) be an arbitrary finite graph. Then either \(\Gamma\)-comparison holds in every metric space or \(\Gamma\)-comparison implies \(C_4\)- or \(T_3\)-comparison.
As a consequence of the above theorem the following is proved:
Corollary. Let \(\Gamma\) be a finite connected graph. Suppose that \(\Gamma\)-comparison is trivial. Then \(\Gamma\) can be constructed from a path \(P_\ell\) of length \(\ell\geq 0\) and two complete graphs \(K_{m_1}\) and \(K_{m_2}\) by attaching \(k_1\) vertices to the left end of \(P_\ell\) and \(k_2\) vertices of \(K_{m_2}\) to the right end of \(P_\ell\).
Reviewer: G. N. Prasanth (Alappuzha)On the geometry of generalized almost quaternionic manifolds of vertical typehttps://zbmath.org/1521.530362023-11-13T18:48:18.785376Z"Arsen'eva, Ol'ga Evgen'evna"https://zbmath.org/authors/?q=ai:arseneva.olga-evgenevnaThe author considers generalized almost quaternionic manifolds of vertical type.
The main result is the following criterion in terms of auto-dual forms for some 4-dimensional manifolds to be Einstein:
Theorem. A 4-dimensional manifold with a Riemannian or neutral pseudo-Riemannian metric is Einstein if and only if its module of auto-dual forms is invariant with respect to the Riemann-Christoffel endomorphism.
The author also considers the connection of this criterion with similar results for 4-dimensional Riemannian manifolds and quaternionic-Kähler manifolds.
Reviewer: Mihail Banaru (Smolensk)Homogeneous Einstein metrics and butterflieshttps://zbmath.org/1521.530372023-11-13T18:48:18.785376Z"Böhm, Christoph"https://zbmath.org/authors/?q=ai:bohm.christoph.1"Kerr, Megan M."https://zbmath.org/authors/?q=ai:kerr.megan-mA Riemannian manifold \((M, g)\) is called Einstein if it has constant Ricci tensor, that is if \(\mathrm{Ric}(g) =\lambda\cdot g\), \(\lambda\in \mathbb{R}\). The present work refers to the homogeneous setting, where a compact homogeneous space \((M, g)\) is a compact Riemannian manifold on which a compact Lie group \(G\) acts transitively by isometries. Then, it has a presentation \(M = G/H\), where \(H\) is a compact subgroup of \(G\). General existence results are difficult to obtain. However, in [\textit{C. Böhm}, J. Differ. Geom. 67, No. 1, 79--165 (2004; Zbl 1098.53039); \textit{C. Böhm} et al., Geom. Funct. Anal. 14, No. 4, 681--733 (2004; Zbl 1068.53029)], the authors reduced the existence problem to the study of Lie algebraic objects, such as simplicial complexes and graphs. On the other hand, \textit{M. M. Graev} [Trans. Mosc. Math. Soc. 2012, 1--28 (2012; Zbl 1278.53043); translation from Tr. Mosk. Mat. O.-va 73, No. 1, 1--35 (2012)], associated with a compact homogeneous space \(G/H\) the nerve \(X_{G/H}\), whose non-contractibility implies the existence of a \(G\)-invariant Einstein metric on \(G/H\). The nerve \(X_{G/H}\) is a compact semialgebraic set defined Lie-theoretically by intermediate subgroups.
The first goal of the present paper is to present a very detailed analysis of Graev's work. The second one is to give a shorter proof of the following first author's result in [J. Differ. Geom. 67, No. 1, 79--165 (2004; Zbl 1098.53039)]: if the simplicial complex \(\Delta _{G/H}\) of a compact homogeneous space with finite fundamental group and \(G, H\) connected, is not contractible, then \(G/H\) admits a \(G\)-invariant Einstein metric.
The paper also reviews some known classification results, and proposes some open problems about homogeneous Einstein metrics.
Reviewer: Andreas Arvanitoyeorgos (Pátra)Negative Sasakian structures on simply-connected \(5\)-manifoldshttps://zbmath.org/1521.530382023-11-13T18:48:18.785376Z"Muñoz, Vicente"https://zbmath.org/authors/?q=ai:munoz.vicente"Schütt, Matthias"https://zbmath.org/authors/?q=ai:schutt.matthias"Tralle, Aleksy"https://zbmath.org/authors/?q=ai:tralle.aleksejThe following questions appear in the well-known seminal book on Sasakian geometry by \textit{C. P. Boyer} and \textit{K. Galicki} [Sasakian geometry. Oxford: Oxford University Press (2008; Zbl 1155.53002)]:
Question 1. Which simply connected rational homology \(5\)-spheres admit negative Sasakian structures?
Question 2. Which simply connected rational homology spheres admit both positive and negative Sasakian structures?
The authors contribute to answer the first question, and give a complete answer to the second question by proving the following:
Theorem. All positive Sasakian simply connected rational homology spheres also admit negative Sasakian structures.
Reviewer: Mircea Crâşmăreanu (Iaşi)The spinor and tensor fields with higher spin on spaces of constant curvaturehttps://zbmath.org/1521.530392023-11-13T18:48:18.785376Z"Homma, Yasushi"https://zbmath.org/authors/?q=ai:homma.yasushi"Tomihisa, Takuma"https://zbmath.org/authors/?q=ai:tomihisa.takumaThe paper under review considers the standard first-order differential operators \(D_j\) on spinor fields with spin \(j+1/2\) over Riemannian spin manifolds. The case \(j=0\) corresponds to the classical Dirac operator, and the Rarita-Schwinger operator occurs for \(j=1\).
After several observations in the general case in Section 2, the authors focus on spaces of constant sectional curvature. They compute for the round sphere the spectra of \(D_j\) and some other related operators, including the standard Laplacian acting on the smooth sections of the mentioned fiber bundle.
The cases of trace-free symmetric tensor fields and spinor fields coupled with differential forms are also considered.
Reviewer: Emilio A. Lauret (Bahía Blanca)Hypercomplex almost abelian solvmanifoldshttps://zbmath.org/1521.530402023-11-13T18:48:18.785376Z"Andrada, Adrián"https://zbmath.org/authors/?q=ai:andrada.adrian"Barberis, María Laura"https://zbmath.org/authors/?q=ai:barberis.maria-lauraA hypercomplex structure on a smooth manifold \(M\) is a triple \(J_1,J_2,J_3\in\Gamma(\mathrm{End}(TM))\) of complex structures satisfying the quaternionic relations. As with many geometric structures it is instructive to study homogeneous examples and more specifically left-invariant structures on Lie groups. In particular, hypercomplex nil- and solvmanifolds (i.e., quotients of nilpotent, or solvable Lie groups by a cocompact lattice) provide an interesting source of examples on the one hand and a testing ground for various conjectures on the other hand.
The paper contributes to the understanding of invariant hypercomplex structures on solvmanifolds by providing a characterization of left-invariant hypercomplex structures on almost abelian Lie groups and associated solvmanifolds and giving a criterion for the existence of a compatible HKT metric. The authors go on to classify the hypercomplex almost abelian Lie groups in dimension \(8\), determine which of them admit lattices and show that in fact any flat compact hyper-Kähler manifold of dimension \(8\) is isometric to such a solvmanifold with an invariant hyper-Kähler structure.
A Lie group \(G\) is called almost abelian if its Lie algebra \(g\) contains an abelian ideal of codimension one. One of the key technical ingredients of the paper is the observation that if \(n=\dim g\), then \(g\) is of the form \(\mathbb{R}e_0\ltimes \mathbb{R}^{n-1}\) and the Lie algebra structure is essentially determined by the matrix \(A\in gl(n-1,\mathbb R)\) corresponding to the linear map \(\mathrm{ad}_{e_0}\). The characterization and classification results are then given in terms of the matrix \(A\).
Reviewer: Markus Röser (Hamburg)Geometric properties of orbits of Hermann actionshttps://zbmath.org/1521.530412023-11-13T18:48:18.785376Z"Ohno, Shinji"https://zbmath.org/authors/?q=ai:ohno.shinjiAuthor's abstract: In this paper, we investigate properties of orbits of Hermann actions as submanifolds without assuming the commutability of involutions which define Hermann actions. In particular, we compute the second fundamental form of orbits of Hermann actions, and give a sufficient condition for orbits of Hermann actions to be weakly reflective submanifolds and arid submanifolds.
Reviewer: Sergei S. Platonov (Petrozavodsk)A variational characterization of calibrated submanifoldshttps://zbmath.org/1521.530422023-11-13T18:48:18.785376Z"Cheng, Da Rong"https://zbmath.org/authors/?q=ai:cheng.da-rong"Karigiannis, Spiro"https://zbmath.org/authors/?q=ai:karigiannis.spiro"Madnick, Jesse"https://zbmath.org/authors/?q=ai:madnick.jesseThe typical setting in calibrated geometries consists of a manifold and a calibration compatible with (or inducing) a Riemannian metric on the manifold, and then possibly the search for the calibrated submanifolds, which are in particular of minimal volume due to the properties of the calibration being both of comass 1 and a closed differential form.
In this article, the authors begin with a Riemannian manifold \((\overline{M},\overline{g})\), consider any given closed embedded \(k\)-submanifold \(M\subset \overline{M}\), and study the volume functional \(\overline{g}\longmapsto\mathcal{V}(\overline{g})= \mathrm{vol}(M,g)= \int_M\mathrm{vol}_{(M,g)}\) where \(g=\overline{g}_{|M}\).
The functional clearly has no critical points on the space of all metrics on \(\overline{M}\), so a novel kind of calibration theory is designed in this paper. The problem was also studied in [\textit{C. Arezzo} and \textit{J. Sun}, J. Reine Angew. Math. 709, 171--200 (2015; Zbl 1331.53105); \textit{J. Sun}, J. Math. Anal. Appl. 434, 1474--1488 (2016; Zbl 1328.53096); \textit{Q. Tan}, Ann. Global Anal. Geom. 53, 217--231 (2018; Zbl 1468.53021)].
The manifold is now assumed to be endowed with a set of \(k\)-forms \(\mu\), not necessarily closed. Then the variational calculus of \(\mathcal V\) is restricted to those metrics which live in a subclass \(\mathcal G\) determined by \(\mu\), where a fortiori it is required that \(\overline{g}\) keeps \(\mu\) with comass 1.
A general and main assertion is the so-called ``meta-theorem''. Roughly, condensed in Theorems A and B in the paper, it says that the metric \(\overline{g}\) on \(\overline{M}\) is a critical point of \(\mathcal V\) if and only if \(M\) is calibrated by \(\mu\), this is, \(\mu_{|M}=\mathrm{vol}_{(M,g)}\). The result is established rigorously and developed \textit{only} in four specific contexts or geometries:
(1) The first case is that of a Hermitian structure \((\overline{M},\overline{g},J,\omega)\). Here, the family \(\mathcal G\) is given as the metrics on \(\overline{M}\) of the form \(\overline{g}_t(X,Y) = \omega_t(X,JY)\), where \(\omega_t =\omega + \mathrm{d}\alpha_t^{(1,1)}\) and \(\alpha_t\) is a one-parameter family of compactly supported smooth 1-forms on \(\overline{M}\) with \(\alpha_{0}=0\), for \(|t|\) sufficiently small so as to assure positive definiteness. All metrics define Hermitian structures (eventually with torsion) compatible with the same \(J\). The \(2k\)-forms \(\mu\) are now the typical forms \(\frac{1}{k!}\omega_t^k\) and the fixed oriented closed \(2k\)-submanifold \(M\) is a complex submanifold of \((\overline{M},J)\).
(2) The second case is a similar variation of a metric induced by a \(\mathrm{G}_2\)-structure \(\varphi\) on a given 7-dimensional manifold \((\overline{M},\varphi)\). As it is well known, the structure 3-form \(\varphi\) induces a metric \(\overline{g}\). Hence the authors may consider small deformations via \(\varphi_t=\varphi+\mathrm{d}\beta_t\) with \(\beta_{0}=0\). The submanifold \(M\) is a 3-manifold, and if calibrated it is associative. Equivalently, \(\overline{g}\) is critical for the functional \(\mathcal V\) in the directions of \(\mathrm{d}\dot{\beta}={\frac{d}{dt}}_{\mid_{t=0}}{\varphi}_t\). Indeed the main theorem holds true for the starting \(\mathrm{G}_2\)-structure.
(3) The third case is that of the calibration \(\star\varphi\) and the corresponding calibrated and coassociative submanifolds.
(4) The fourth case is for \(\mathrm{Spin}(7)\) structures on 8-manifolds and the Cayley submanifolds. Here the family of variations of the metric has to be carefully chosen and theorems A and B reveal marked differences from the first three cases.
A difficult task in all four contexts seems to be proving Theorem B, i.e., to see that \(M\) is calibrated by \(\mu\) if \(\overline{g}\) is a critical point of \(\mathcal V\), and moreover to see that \(M\) is of the required type, respectively, complex, associative, coassociative or Cayley.
The last section is devoted to observations and generalisations of the variational characterization of submanifolds. Firstly, the more abstract proof of Theorem A in all cases, i.e., to see that \(\overline{g}\) is a critical point of \(\mathcal V\) if \(M\) is calibrated by \(\mu\). Secondly, considering the possible role of the Berger-Ebin theorem in the study of such variations. Thirdly, the framework of variations of the metric on \(\overline{M}\) induced by the \(k\)-forms \(f_t^*\mu\) defined by the flow of diffeomorphisms \(f_t\) of a given vector field on \(\overline{M}\). Fourthly, an application of the theory to the variational characterization of the so-called Smith maps, providing further a kind of Theorems A and B about the equivalence between such maps and critical points of the \(k\)-energy.
Reviewer: Rui Albuquerque (Lisboa)Classification of rank-one submanifoldshttps://zbmath.org/1521.530432023-11-13T18:48:18.785376Z"Raffaelli, Matteo"https://zbmath.org/authors/?q=ai:raffaelli.matteoSummary: We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold \(\sigma \), we associate an integer-valued function, called \textit{degree}, measuring the extent to which \(\sigma\) fails to be cylindrical. In particular, we show that if the degree is constant and equal to \(d\), then the singularities of \(\sigma\) can only occur along an \((m-d)\)-dimensional ``striction'' submanifold. This result allows us to extend the standard classification of developable surfaces in \(\mathbb{R}^3\) to the whole family of flat and ruled submanifolds without planar points, also known as \textit{rank-one}: an open and dense subset of every rank-one submanifold is the union of \textit{cylindrical}, \textit{conical}, and \textit{tangent} regions.Diameter estimates for submanifolds in manifolds with nonnegative curvaturehttps://zbmath.org/1521.530442023-11-13T18:48:18.785376Z"Wu, Jia-Yong"https://zbmath.org/authors/?q=ai:wu.jiayongSummary: Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of \textit{P. Topping} [Comment. Math. Helv. 83, No. 3, 539--546 (2008; Zbl 1154.53007)], \textit{J.-Y. Wu} and \textit{Y. Zheng} [Proc. Am. Math. Soc. 139, No. 11, 4097--4104 (2011; Zbl 1230.53054)] and \textit{S.-H. Paeng} [Differ. Geom. Appl. 33, 127--138 (2014; Zbl 1325.53072)].Harmonic metrics, harmonic tensors and their applicationshttps://zbmath.org/1521.530452023-11-13T18:48:18.785376Z"Chen, Bang-Yen"https://zbmath.org/authors/?q=ai:chen.bang-yenIn this survey paper, the author summarizes the main results from the mathematical literature concerning the harmonicity of metrics and tensors. The research in this field was initiated in [the author and \textit{T. Nagano}, J. Math. Soc. Japan 36, 295--313 (1984; Zbl 0543.58015)], where a harmonic metric was defined as a metric \(g'\) on a Riemannian manifold \((M, g)\), such that the identity map \(\mathrm{id}_M : (M, g) \rightarrow (M,g')\) is harmonic. For the harmonicity of a metric \(g'\) on a Riemannian manifold \((M,g)\) the author obtains firstly a characterization in terms of the Christoffel symbols with respect to a local coordinate system of \(M\), namely
\[
g^{ji}\left({\Gamma'}_{ji}^k-\Gamma_{ji}^k\right)=0,
\]
and secondly a characterization which involves the function \(f := \mathrm{ tr}_gg'\) and a vector field \(\omega\) with components \(\omega_j=\nabla^kg'_{kj}\) satisfying the relation
\[\nabla_jf=2\omega_j. \]
A symmetric tensor field of type \((0,2)\) on a Riemannian manifold \((M,g)\) satisfying such a condition is called a harmonic tensor field. The author shows that the space of all harmonic symmetric tensors of type \((0,2)\) on a Riemannian manifold \((M,g)\) of dimension \(n\) is isomorphic to the space of all conservative symmetric tensors of type \((0,2)\) on \((M,g)\) if \(n\geq 3\), and it decomposes into the direct sum
\[ \{\lambda g,\ \lambda \text{ smooth function on }M\}\oplus\{T\in\ker(\delta): \mathrm{tr} T=0\} \]
if \(n=2\). Then the author proves that for any Riemannian manifold \((M,g)\) of dimension \(\geq 2\) the Ricci tensor is harmonic with respect to \(g\) and the conservativity of the Ricci tensor is equivalent to the property of \((M,g)\) having constant scalar curvature. Moreover, if \((M,g)\) is an orientable 2-dimensional Riemannian manifold, then the space \(\{T \in \ker(\delta) : \mathrm{tr}_g T = 0\}\) is linearly isomorphic to the space of holomorphic quadratic differentials on \((M, g)\) endowed with the natural complex structure. If \(M\) is diffeomorphic either to a 2-sphere or to a real projective plane, then all harmonic metrics on \((M, g)\) are conformal changes of the metric \(g\).
The author shows that the nullity of the identity map of a Riemannian manifold is equal to the dimension of the space
\[\ker(\delta)^\perp = \{T \in \mathcal{S}: T = \mathcal{L}_vg\text{ for some geodesic vector field }v\}.\]
The notion of geodesic vector field on a Riemannian manifold \((M,g)\) was defined by \textit{K. Yano} and \textit{T. Nagano} in [C. R. Acad. Sci. Paris 252, 504--505, (1961; Zbl 0100.35903)] as a vector field \(v=(v^i)\) satisfying the condition \[ g^{ji}\Delta_j\Delta_iv^k +\mathrm{Ric}^k_iv^i = 0. \] In [\textit{K. Yano} and \textit{T. Nagano}, ``On geodesic vector fields in a compact orientable Riemannian space'', Comm. Math. Helv. 35, 55--64 (1961)] the authors proved the nonexistence of nonzero geodesic vector fields on a compact Riemannian manifold with negative-definite Ricci tensor. They showed that on an Einstein manifold \((M, g)\) with \(\mathrm{Ric} = cg\) the divergence of a geodesic vector field is a solution of the equation \(\Delta f = 2cf\) and a geodesic vector field is decomposed uniquely as the sum of a Killing vector field and a solution of the equation \(\Delta f = 2cf\).
In [the author and Nagano, loc. cit.] it was proved that the geodesic vector fields on a Riemannian manifold \((M, g)\) are the infinitesimal generators of the harmonic metrics on \((M, g)\). The authors showed that on a compact Kähler manifold \((M, J, g)\) a vector field \(v\) is a holomorphic vector field if and only if \(g'= \mathcal{L}_vg\) is a harmonic tensor on \((M, J, g)\). Moreover, they proved that if \(g'\) is a conservative harmonic metric of \((M, g)\) such that \(\mathrm{tr}_gg'=\mathrm{tr}_gg\), then the identity map \(i_M : (M, g) \rightarrow (M, g')\) is volume-decreasing, i.e., \(dv_g'\leq dv_g\) at each point of \(M\), and \(dv_{g'} = dv_g\) if and only if \(g' = g\) on \(M\).
For harmonic metrics on Kähler and hyper-Kähler manifolds the author quotes some results proved by \textit{Y. Watanabe} and \textit{R. Dohira} in [Math. J. Toyama Univ. 18, 137--146 (1995; Zbl 0848.53013)] and by \textit{A. D. Vîlcu} and \textit{G. E. Vîlcu} in [JP J. Geom. Topol. 7, 397--403 (2007; Zbl 1136.53318)].
In the previously cited paper by the author and Nagano [loc. cit.], it was shown that an isometric immersion \(\phi : (M, g_0) \rightarrow(E^m, \bar{g}_0)\) of a surface into a Euclidean space is harmonic if and only if \(\phi : (M, G_0) \rightarrow(E^m, \bar{g}_0)\) is harmonic, where \(G_0\) is the metric induced on \(M\) by the Gauss map \(\nu\) associated with \(\phi\). Subsequently, the Gauss map \(\nu\) is harmonic and the identity map \(\mathrm{id}_M:(M,G_0)\rightarrow (M,g_0)\) is homothetic if and only if \((M, g_0)\) has constant Gauss curvature or equivalently, there exists a hypersphere \(S^{m-1}\) of \(E^m\) containing \(\phi(M)\) such that \(\phi:(M, g_0) \rightarrow S^{m-1}\) is a harmonic map. This condition is equivalent to the harmonicity of the map \(\phi : (M, G_0) \rightarrow (E^{n+1}, \bar{g}_0)\) and it characterizes the situation when the identity map is conformal and the Gauss map is harmonic. If the identity map is an affine map, then either it is homothetic or the surfaces \((M, g_0)\) and \((M, G_0)\) are flat. Moreover, the identity map is an affine map and the Gauss map associated with \(\phi\) is harmonic if and only if \((M, g_0)\) has constant Gauss curvature and either \((M, g_0)\) is immersed in a hypersphere of \(E^m\) as a minimal surface via \(\phi\), or \((M, g_0)\) is immersed as an open subset of the product surface of two planar circles via \(\phi\).
In the book [the author and \textit{L. Verstraelen}, Laplace transformations of submanifolds. Leuven, Belgium: Katholieke Universiteit (1995; Zbl 0912.53036)], the authors considered an isometric immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\) of a Riemannian \(n\)-manifold into Euclidean \(m\)-space and studied the conditions under which the Laplace transformation \(\mathrm{id}_M:(M,g_0)\rightarrow(M, g_{L_\phi})\) determines the immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\), where \(L_\phi\) is the Laplace map and \(g_{L_\phi}\) is the metric induced via \(L_\phi\). The main results of this book are resumed in a section of the paper under review. Another section is devoted to the results on the Laplace-Gauss identity map (or LG-identity map) obtained in the same book. The property of the LG-identity map to be conformal or homothetic is related to the geometry of the manifold \(M\) and to the properties of the isometric immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\).
Regarding the tangent bundle of a pseudo-Riemannian manifold \((M,g)\), the author recalls that the identity map \(\mathrm{id}_{TM} : (TM, G) \rightarrow(TM, g^c)\) is biharmonic (i.e., the complete lift \(g^c\) of the metric \(g\) is biharmonic with respect to the Sasaki metric \(G\)) and that it is totally geodesic if and only if the projection \(\pi:(TM,G)\rightarrow(M,g)\) has the same property (see [\textit{C. Oniciuc}, Bull. Belg. Math. Soc. Simon Stevin 7, 443--454 (2000; Zbl 0983.53042)]). The author also recalls that in the more general case when \(G\) is an arbitrary \(g\)-natural metric on \(TM\), the identity map \(\mathrm{id}_{TM} : (TM, g^S) \rightarrow(TM,G)\) is harmonic if and only if a certain relation between some functions appearing in the expression of the metric \(G\) is satisfied and either the horizontal and vertical distributions are orthogonal with respect to \(G\), or \((M,g)\) is an Einstein manifold and a relation between two functions involved in the expression of \(G\) holds. Another characterization for the harmonicity of \(\mathrm{id}_{TM}\) consists in two relations satisfied by the functions from the expression of \(G\) (see [\textit{M.T.K. Abbassi} and \textit{G. Calvaruso}, Rend. Semin. Mat. Univ. Politec. Torino 68, 37--56 (2010; Zbl 1202.58011)]). For the case when \(g\) and \(\hat{g}\) are \(G\)-invariant pseudo-Riemannian metrics on a nonreductive homogeneous 4-manifold \(M^4\), it was proved in [\textit{A. Zaeim} and \textit{P. Atashpeykar}, Czechoslovak Math. J. 68(143), 475--490 (2018; Zbl 1488.53192)] that a pseudo-Riemannian metric \(\hat{g}\) is a harmonic metric on \((M^4, g)\) and the identity map of \(TM^4\) endowed with Sasaki metric, horizontal and complete lifts of the metrics \(g\) and \(\hat{g}\) is harmonic. In [\textit{C. L. Bejan} and \textit{S. L. Druță-Romaniuc}, Mediterr. J. Math. 12, 481--496 (2015; Zbl 1322.53065)] the authors proved the equivalence between the harmonicity of a Walker metric \(\hat{g}\) on a 4-dimensional Walker manifold \((W^4,g,\mathcal{D})\) and the harmonicity of the Sasaki metric \(\hat{g}^S\) and of the horizontal lift \(\hat{g}^h\) with respect to \({g}^S\) and \({g}^h\), respectively. Similar results for the harmonicity of Gödel-type metrics on Gödel-type spacetimes were obtained in [\textit{A. Zaeim} et al., Int. J. Geom. Methods Mod. Phys. 17, Article 2050092, 16 p. (2020)].
A (relative) biharmonic metric with respect to another metric \(g\) was defined in [\textit{P. Baird} and \textit{D. Kamissoko}, Ann. Global Anal. Geom. 23, 65--75 (2003; Zbl 1027.31004)] as a metric \(\bar{g}\) for which the identity map \(\mathrm{id}_M:(M, g) \rightarrow (M, \bar{g})\) is biharmonic. In the paper under review the author mentions some examples of proper biharmonic metrics constructed in his joint work with \textit{Y.-L. Ou} [Biharmonic submanifolds and biharmonic maps in Riemannian geometry, Hackensack, NJ: World Scientific (2020; Zbl 1455.53002)]. He also recalls that on a compact Einstein \(m\)-manifold \((M, g)\), for which \(\mathrm{Ric} = c g\), with \(m \geq 6\) and \(c < 0\), or with \(m > 6\) and \(c \leq 0\), the metric \(g' = e^{2\rho}g\) is biharmonic if and only if it is a harmonic metric for \((M, g)\).
The author quotes some references for harmonic metrics and compact symmetric spaces, stability of other spaces, Laplace transformations of submanifolds, harmonic metrics on generalized Sasakian space forms and on light-like manifolds or semi-Riemannian manifolds.
At the end of the paper, the author defines the notion of \(p\)-harmonic metric on a Riemannian manifold \((M,g)\) in an analogous way as for harmonic and biharmonic metrics, by the \(p\)-harmonicity of the identity map \(\mathrm{id}_M : (M, g) \rightarrow (M, g')\) and he proposes some open problems concerning the relations between \(p\)-harmonic metrics and \(k\)-harmonic metrics on a Riemannian manifold \((M, g)\) for \(k < p\).
For the entire collection see [Zbl 1490.53003].
Reviewer: Simona Druta-Romaniuc (Iaşi)Infinite energy harmonic maps from Riemann surfaces to CAT(0) spaceshttps://zbmath.org/1521.530462023-11-13T18:48:18.785376Z"Daskalopoulos, Georgios"https://zbmath.org/authors/?q=ai:daskalopoulos.georgios-d"Mese, Chikako"https://zbmath.org/authors/?q=ai:mese.chikakoSummary: We present results about harmonic maps with possibly infinite energy from punctured Riemann surfaces to CAT(0) spaces. In particular, we give precise estimates of their energy growth near the punctures and prove uniqueness.Triharmonic curves in Bianchi-Cartan-Vranceanu spaceshttps://zbmath.org/1521.530472023-11-13T18:48:18.785376Z"Senoussi, Bendehiba"https://zbmath.org/authors/?q=ai:senoussi.bendehibaThe Bianchi-Cartan-Vranceanu spaces form a \(2\)-parameter family of \(3\)-dimensional homogeneous Riemannian manifolds with \(6\)- and \(4\)-dimensional isometry group. The paper provides a characterization of triharmonic curves in these spaces based on curvature and torsion functions of the given curve. The case of helices is studied in detail.
Reviewer: Mircea Crâşmăreanu (Iaşi)Equidistant sets on Alexandrov surfaceshttps://zbmath.org/1521.530482023-11-13T18:48:18.785376Z"Fox, Logan S."https://zbmath.org/authors/?q=ai:fox.logan-s"Veerman, J. J. P."https://zbmath.org/authors/?q=ai:veerman.j-j-pSummary: We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditionshttps://zbmath.org/1521.530492023-11-13T18:48:18.785376Z"Braun, Mathias"https://zbmath.org/authors/?q=ai:braun.mathiasSummary: For a Lorentzian space measured by \(\mathfrak{m}\) in the sense of [\textit{M. Kunzinger} and \textit{C. Sämann}, Ann. Global Anal. Geom. 54, No. 3, 399--447 (2018; Zbl 1501.53057); \textit{F. Cavalletti} and \textit{A. Mondino}, Gen. Relativ. Gravitation 54, No. 11, Paper No. 137, 39 p. (2022; Zbl 1518.83004)], we introduce and study synthetic notions of timelike lower Ricci curvature bounds by \(K\in\mathbb{R}\) and upper dimension bounds by \(N\in[1,\infty)\), namely the timelike curvature-dimension conditions \(\mathrm{TCD}_p(K,N)\) and \(\mathrm{TCD}_p^\ast(K,N)\) in weak and strong forms, where \(p\in(0,1)\), and the timelike measure-contraction properties \(\mathrm{TMCP}(K,N)\) and \(\mathrm{TMCP}^\ast(K,N)\). These are formulated by convexity properties of the Rényi entropy with respect to \(\mathfrak{m}\) along \(\ell_p\)-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological \(\ell_p\)-optimal couplings and chronological \(\ell_p\)-geodesics. We also prove the equivalence of \(\mathrm{TCD}_p^\ast(K,N)\) and \(\mathrm{TMCP}^\ast(K,N)\) to their respective entropic counterparts in the sense of [Cavalletti and Mondino, loc. cit.]. Some of these results are obtained under timelike \(p\)-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.Calabi-Yau metrics with conical singularities along line arrangementshttps://zbmath.org/1521.530502023-11-13T18:48:18.785376Z"de Borbon, Martin"https://zbmath.org/authors/?q=ai:de-borbon.martin"Spotti, Cristiano"https://zbmath.org/authors/?q=ai:spotti.cristianoConical Kähler-Einstein metrics with singularities along a divisor became relevant in Kähler geometry since the seminal work of \textit{X. Chen} et al. [J. Am. Math. Soc. 28, No. 1, 183--197 (2015; Zbl 1312.53096)]. It is natural to ask what happens when the divisor becomes singular, and this paper deals with one of the simplest of such cases, namely when the singularity occurs along a line arrangement in the 2-dimensional complex projective space, with conical angles prescribed by weight data, and the local behavior near the intersections of the divisors for modeled on polyhedral Kähler cone metrics. The main result is an existence result of a Calabi-Yau metric with the prescribed singularity, based on an adaption of Yau's continuity path, and some extra linear theory. As a partially conjectural application, the authors discuss a log version of the Bogomolov-Miyaoka-Yau inequality.
Reviewer: Yang Li (Cambridge, MA)Families of almost complex structures and transverse \((p, p)\)-formshttps://zbmath.org/1521.530512023-11-13T18:48:18.785376Z"Hind, Richard"https://zbmath.org/authors/?q=ai:hind.richard"Medori, Costantino"https://zbmath.org/authors/?q=ai:medori.costantino"Tomassini, Adriano"https://zbmath.org/authors/?q=ai:tomassini.adrianoSummary: An \textit{almost \(p\)-Kähler manifold} is a triple \((M,J,\Omega)\), where \((M, J)\) is an almost complex manifold of real dimension \(2n\) and \(\Omega\) is a closed real transverse \((p, p)\)-form on \((M, J)\), where \(1\le p\le n\). When \(J\) is integrable, almost \(p\)-Kähler manifolds are called \(p\)-\textit{Kähler manifolds}. We produce families of almost \(p\)-Kähler structures \((J_t,\Omega_t)\) on \(\mathbb{C}^3, \mathbb{C}^4\), and on the real torus \(\mathbb{T}^6\), arising as deformations of Kähler structures \((J_0,g_0,\omega_0)\), such that the almost complex structures \(J_t\) cannot be locally compatible with any symplectic form for \(t\ne 0\). Furthermore, examples of special compact nilmanifolds with and without almost \(p\)-Kähler structures are presented.Cohomogeneity one central Kähler metrics in dimension fourhttps://zbmath.org/1521.530522023-11-13T18:48:18.785376Z"Jeffres, Thalia"https://zbmath.org/authors/?q=ai:jeffres.thalia-d"Maschler, Gideon"https://zbmath.org/authors/?q=ai:maschler.gideon"Ream, Robert"https://zbmath.org/authors/?q=ai:ream.robertSummary: A Kähler metric is called central if the determinant of its Ricci endomorphism is constant; see [\textit{G. Maschler}, Trans. Am. Math. Soc. 355, No. 6, 2161--2182 (2003; Zbl 1043.53056)]. For the case in which this constant is zero, we study on 4-manifolds the existence of complete metrics of this type which have cohomogeneity one for three unimodular 3-dimensional Lie groups: SU(2), the group \(E(2)\) of Euclidean plane motions, and a quotient by a discrete subgroup of the Heisenberg group \(\mathrm{nil}_3\). We obtain a complete classification for SU(2), and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.On the existence of mass minimizing rectifiable \(G\) chains in finite dimensional normed spaceshttps://zbmath.org/1521.530532023-11-13T18:48:18.785376Z"De Pauw, Thierry"https://zbmath.org/authors/?q=ai:de-pauw.thierry"Vasilyev, Ioann"https://zbmath.org/authors/?q=ai:vasilyev.ioannSummary: We introduce the notion of density contractor of dimension \(m\) in a finite dimensional normed space \(X\). If \(m+1=\dim X\), this includes the area contracting projectors on hyperplanes whose existence was established by \textit{H. Buseman} [Ann. Math. (2) 48, 234--267 (1947; Zbl 0029.35301)]. If \(m=2\), density contractors are an ersatz for such projectors and their existence, established here, follows from works by \textit{D. Burago} and \textit{S. Ivanov} [Geom. Funct. Anal. 14, No. 3, 469--490 (2004; Zbl 1067.53042); Geom. Funct. Anal. 22, No. 3, 627--638 (2012; Zbl 1266.52008)]. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable \(G\) chains, regardless of the group of coefficients \(G\). This is obtained as a consequence of the lower semicontinuity of the \(m\) dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.On \(\delta\)-Casorati curvature invariants of Lagrangian submanifolds in quaternionic Kähler manifolds of constant \(q\)-sectional curvaturehttps://zbmath.org/1521.530552023-11-13T18:48:18.785376Z"Aquib, Mohd"https://zbmath.org/authors/?q=ai:aquib.mohd"Lone, Mohamd Saleem"https://zbmath.org/authors/?q=ai:lone.mohamd-saleem"Neacşu, Crina"https://zbmath.org/authors/?q=ai:neacsu.crina"Vîlcu, Gabriel-Eduard"https://zbmath.org/authors/?q=ai:vilcu.gabriel-eduardThis article is about \(\delta\)-Casorati curvature invariants of Lagrangian submanifolds of quaternionic space forms. Let us explain these terms one by one, in reverse order. Let \((M,g,\mathcal Q)\) be a quaternionic Kähler manifold, where \(\mathcal Q\subset \mathrm{End}\,(TM)\) is the quaternionic structure bundle. A tangent 2-plane \(\Pi\subset T_xM\) is called half-quaternionic if it admits a basis \(\{U_1,U_2\}\) such that \(\mathcal Q(U_1)=\mathcal Q(U_2)\). The sectional curvature of a half-quaternionic 2-plane is called a \(q\)-sectional curvature, and a manifold of constant \(q\)-sectional curvature is known as a quaternionic space form. The simply connected quaternionic space forms are \(\mathbb H\mathrm{P}^n\) (positive \(q\)-sectional curvature), \(\mathbb{H}\mathrm{H}^n\) (negative) and \(\mathbb{H}^n\) (zero). A submanifold \(X\subset M\) is called totally real if its tangent spaces are mapped by \(\mathcal Q\) into the corresponding normal spaces, and a totally real submanifold of maximal dimension (one quarter of the ambient space) is called Lagrangian.
The Casorati curvature of a submanifold is simply the norm of its second fundamental form. This extrinsic invariant of the submanifold is related, via inequalities first introduced by B.~Y.~Chen, to intrinsic invariants built out of the scalar curvature and the pointwise infimum of the sectional curvature of the submanifold, called \(\delta\)-invariants. Chen's inequalities and their generalizations are of fundamental importance in the theory of submanifolds of space forms. The Casorati curvature can also be modified by adding a term proportional to its pointwise infimum (or supremum) over all hyperplanes in the tangent space to obtain (generalized) \(\delta\)-Casorati curvature invariants.
With all this in mind, the main result of this rather technical paper consists of two Chen-type inequalities relating the scalar curvature of a Lagrangian submanifold in a quaternionic space form to its generalized \(\delta\)-Casorati curvature and Casorati curvature. The proof is a lengthy computation. The remainder of the paper is spent characterizing the equality cases of these two inequalities, and listing known examples. The authors prove that such submanifolds must be totally geodesic if the first inequality is saturated, and either totally geodesic or so-called Lagrangian \(H\)-umbilical in case the second inequality is saturated.
Reviewer: Daniel Thung (Amsterdam)Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic typehttps://zbmath.org/1521.530562023-11-13T18:48:18.785376Z"Goto, Ryushi"https://zbmath.org/authors/?q=ai:goto.ryushiThe main objectives of this article are to formulate and prove analogues of the Matsushima-Lichnerowicz theorem in the setting of generalised geometry, and to construct nontrivial examples of generalised Kähler structures of constant scalar curvature on compact manifolds by deforming Kähler structures of constant scalar curvature.
Recall that the classical Matsushima-Lichnerowicz theorem in the setting of Kähler geometry states that a necessary condition for a compact complex manifold \(X\) to admit Kähler metrics \(\omega\) of constant scalar curvature is that its automorphism group is a reductive Lie group, whose maximal compact subgroup is in fact the group of isometries of the Kähler manifold \((X,\omega)\). This is a rather powerful obstruction that can be used to rule out the existence of Kähler metrics of constant scalar curvature on, for instance, the blowup of the complex projective plane \(\mathbb C\mathbb P^2\) at a point, since its automorphism group is not reductive.
In the context of generalised Kähler geometry, there is no obvious definition of scalar curvature. The key idea is to interpret the scalar curvature as a moment map on a certain infinite-dimensional symplectic manifold. In ordinary Kähler geometry, the infinite-dimensional manifold \(\mathscr M\) in question is the space of all Kähler structures \(J\) on a compact manifold \(X\) compatible with a fixed Kähler form \(\omega\) (which in particular determines a volume form). This infinite-dimensional manifold carries a natural symplectic structure given by
\[\Omega(\dot J_1, \dot J_2) = \int_X \omega(\dot J_1\cdot,\dot J_2\cdot) \wedge \frac{\omega^{n-1}}{(n-1)!}.
\]
Infinitesimal automorphisms of the Kähler manifold induce Hamiltonian vector fields on \(\mathscr M\). The scalar curvature is then a moment map for the action of this automorphism group.
In generalised geometry, one replaces the tangent bundle \(TX\) with the generalised tangent bundle \(\mathbb T X :=TX \oplus T^*X\), which is equipped with a natural inner product. In particular, metrics, \(2\)-forms, and endomorphisms of the tangent bundle can all be regarded as endomorphisms of the generalised tangent bundle \(\mathbb T X\), which suggests that the replacement for a Kähler structure is a pair of endomorphisms \((\mathscr J, \mathscr J_\psi)\) of \(\mathbb T X\) subject to certain compatibility and integrability conditions; see [\textit{M. Gualtieri}, Ann. Math. (2) 174, No. 1, 75--123 (2011; Zbl 1235.32020)]. For a certain class of such structures, known as generalised Kähler structures of symplectic type, the moment map picture in the ordinary Kähler setting generalises and we obtain a notion of scalar curvature for generalised Kähler structures of symplectic type.
The automorphism group of a generalised Kähler manifold is meanwhile extended to include the so-called B-field transformations, which together with the usual diffeomorphisms indeed form a Lie group with Lie bracket on infinitesimal generators given by the Courant bracket. The main result of the paper is that a compact generalised complex manifold \((X,\mathscr J)\) with \(H^1(X;\mathbb C)=0\) admits a generalised Kähler structure \((\mathscr J,\mathscr J_{\psi})\) of constant scalar curvature in the sense described above only if the (extended) automorphism group is a reductive Lie group. If we furthermore assume that the reduced Lie algebra of the automorphism group is trivial, then a compact generalised Kähler manifold \((X,\mathscr J,\mathscr J_{\psi})\) with constant scalar curvature admits deformations with constant scalar curvature. As an explicit application of these results, generalised Kähler structures on \(\mathbb C\mathbb P^2\) with constant scalar curvature are classified in terms of the corresponding holomorphic Poisson structure (which encodes generalised Kähler deformations of the standard Kähler structure).
Reviewer: Arpan Saha (Madrid)Palatini variation in generalized geometry and string effective actionshttps://zbmath.org/1521.530572023-11-13T18:48:18.785376Z"Jurčo, Branislav"https://zbmath.org/authors/?q=ai:jurco.branislav"Moučka, Filip"https://zbmath.org/authors/?q=ai:moucka.filip"Vysoký, Jan"https://zbmath.org/authors/?q=ai:vysoky.janThe authors develop the Palatini formalism within the framework of a generalized Riemannian geometry of Courant algebroids. In the approach known as the Palatini variation, the action corresponding to field equations is a spacetime integral of the scalar curvature. In this context, one had to consider a general torsion-free affine connection. Let \(M\) be a Riemannian manifold with a metric \(g\). First, the authors introduce the required tools. Then, they recall the notions of a Courant algebroid, generalized metrics and courant algebroid connections. They show how ordinary affine connections can naturally be generalizd to Courant algebroid connections. Then, they extend the Palatini method to the setting of the generalized Riemannian geometry of Courant algebroids. The proof of the main results is given in detail and examples are also discussed.
Reviewer: Angela Gammella-Mathieu (Metz)Global uniqueness of the minimal sphere in the Atiyah-Hitchin manifoldhttps://zbmath.org/1521.530592023-11-13T18:48:18.785376Z"Tsai, Chung-Jun"https://zbmath.org/authors/?q=ai:tsai.chung-jun"Wang, Mu-Tao"https://zbmath.org/authors/?q=ai:wang.mu-taoThe authors study the submanifold geometry of the Atiyah-Hitchin manifold, which is a double cover of the 2-monopole moduli space. It should be pointed out that this manifold plays an important role in the supersymmetric background of string theory.
When the manifold is naturally identified as the total space of a line bundle over \(S^2\), the zero section is a distinguished minimal 2-sphere. Concerning the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces, the authors show that this minimal 2-sphere satisfies a certain strong stability condition. The global uniqueness follows as a corollary.
Reviewer: Adrian Sandovici (Piatra Neamt)Coisotropic Hofer-Zehnder capacities of convex domains and related resultshttps://zbmath.org/1521.530622023-11-13T18:48:18.785376Z"Jin, Rongrong"https://zbmath.org/authors/?q=ai:jin.rongrong"Lu, Guangcun"https://zbmath.org/authors/?q=ai:lu.guangcunSummary: We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by \textit{S. Lisi} and \textit{A. Rieser} [J. Symplectic Geom. 18, No. 3, 819--865 (2020; Zbl 1478.53126)] recently), study their estimates and relations with the Hofer-Zehnder capacity, give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by \textit{S. Artstein-Avidan} and \textit{Y. Ostrover} [Int. Math. Res. Not. 2008, Article ID rnn044, 31 p. (2008; Zbl 1149.52006)] and a theorem by \textit{E. Neduv} [Math. Z. 236, No. 1, 99--112 (2001; Zbl 0967.37031)].An \(\varepsilon \)-regularity theorem for line bundle mean curvature flowhttps://zbmath.org/1521.530672023-11-13T18:48:18.785376Z"Han, Xiaoli"https://zbmath.org/authors/?q=ai:han.xiaoli"Yamamoto, Hikaru"https://zbmath.org/authors/?q=ai:yamamoto.hikaruThe line bundle mean curvature flow was introduced by \textit{A. Jacob} and \textit{S.-T. Yau} [Math. Ann. 369, No. 1--2, 869--898 (2017; Zbl 1375.32045)] as the negative gradient flow of a volume functional on the space of Hermitian metrics on a holomorphic line bundle \(L\to X\) over a Kähler manifold \((X,g)\). As the minimizers of the volume functional are deformed Hermitian Yang-Mills (dHYM) metrics, the line bundle mean curvature flow can be used to find the dHYM metrics. Some sufficient conditions were provided in [loc. cit.] to ensure that a line bundle mean curvature flow \(h_t\) defined for \(t\in [0,T)\) can be extended beyond the time \(T\).
In this paper, the authors prove an \(\varepsilon\)-regularity theorem for the line bundle mean curvature flow. This is motivated by the work of \textit{B. White} [Ann. Math. (2) 161, No. 3, 1487--1519 (2005; Zbl 1091.53045)] on the \(\varepsilon\)-regularity of mean curvature flows. For the proof, the authors discover a monotonicity formula and introduce a Gaussian density for line bundle mean curvature flows. This is in analogy with a result in [\textit{G. Huisken}, J. Differ. Geom. 31, No. 1, 285--299 (1990; Zbl 0694.53005)] for mean curvature flows. The authors also introduce the definition of self-shrinkers for line bundle mean curvature flows and prove a Liouville-type theorem, which is used crucially in the proof of the \(\varepsilon\)-regularity theorem.
Reviewer: Yong Wei (Hefei)Corrigendum to: ``On the \(1/H\)-flow by \(p\)-Laplace approximation: new estimates via fake distances under Ricci lower boundshttps://zbmath.org/1521.530682023-11-13T18:48:18.785376Z"Mari, Luciano"https://zbmath.org/authors/?q=ai:mari.luciano"Rigoli, Marco"https://zbmath.org/authors/?q=ai:rigoli.marco"Setti, Alberto G."https://zbmath.org/authors/?q=ai:setti.alberto-gSummary: We correct a mistake in our proof of Lemma 2.17 in our paper [ibid. 144, No. 3, 779--849 (2022; Zbl 1506.53100)]. Although we have to strengthen the assumptions therein and, accordingly, in Theorem 2.22, all of the results on the existence and properties of the IMCF are not affected. Minor changes, with no influence elsewhere in the paper, regard Lemma 3.3, Proposition 4.3 and Lemma 5.3Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski spacehttps://zbmath.org/1521.530692023-11-13T18:48:18.785376Z"Silva Neto, Gregório"https://zbmath.org/authors/?q=ai:silva-neto.gregorio"Silva, Vanessa"https://zbmath.org/authors/?q=ai:silva.vanessa-lSummary: In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.Propagation of symmetries for Ricci shrinkershttps://zbmath.org/1521.530722023-11-13T18:48:18.785376Z"Colding, Tobias Holck"https://zbmath.org/authors/?q=ai:colding.tobias-holck"Minicozzi, William P. II"https://zbmath.org/authors/?q=ai:minicozzi.william-p-iiIn this paper the authors show that if a gradient shrinking Ricci soliton (shrinker) has an approximate symmetry on one scale, then this approximate symmetry propagates outwards to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons. This kind of propagation of symmetry often plays an important role in understanding the structure of solutions, as well as the rate of convergence of a Ricci flow to a singularity.
Reviewer: Abimbola Abolarinwa (Lagos)Dynamical (in)stability of Ricci-flat ALE metrics along the Ricci flowhttps://zbmath.org/1521.530732023-11-13T18:48:18.785376Z"Deruelle, Alix"https://zbmath.org/authors/?q=ai:deruelle.alix"Ozuch, Tristan"https://zbmath.org/authors/?q=ai:ozuch.tristanThe present paper concerns Ricci-flat ALE metrics in Riemannian geometry. Such spaces model the formation of singularities of spaces with Ricci curvature bounds, among other interesting geometrical properties. The authors study the dynamical stability and instability of these spaces along the Ricci flow, using a functional on suitable neighborhoods of any ALE Ricci-flat metrics which detects Ricci-flat metrics as its critical points, that was introduced in their work [``A Łojasiewicz inequality for ALE metrics'', Preprint, \url{arXiv:2007.09937}].
Reviewer: Andreas Arvanitoyeorgos (Pátra)Regularity and curvature estimate for List's flow in four dimensionhttps://zbmath.org/1521.530742023-11-13T18:48:18.785376Z"Wu, Guoqiang"https://zbmath.org/authors/?q=ai:wu.guoqiangThis paper is devoted to List's flow, which is a triple $(M,g(t),\varphi (t))_{t\in (0,T)}$ satisfying
\begin{align*}
\partial_{t}g(t) &=-2\text{Ric}(g(t))+2d\varphi (t)\otimes d\varphi (t), \\
\partial_{t}\varphi (t) &=\Delta_{g(t)}\varphi (t), \\
g(0) &=g_{0},\quad\varphi (0)=\varphi_{0},
\end{align*}
where $\varphi (t):M\rightarrow \mathbb{R}$ are smooth functions, $g_{0}$ is a fixed Riemannian metric, and $\varphi_{0}$ is a fixed smooth function on a compact $n$-dimensional Riemannian manifold $M$ without boundary. Before the work by \textit{B. List} [Commun. Anal. Geom. 16, No. 5, 1007--1048 (2008; Zbl 1166.53044)], the Ricci flow system for a Riemannian metric $\partial_{t}g=-2\mathrm{Ric}(g)$ has been used with great
success for the construction of canonical metrics on Riemannian manifolds of low dimension. B. List has developed a corresponding theory for canonical objects with a certain physical interpretation. He proved the existence of an entropy $E$ such that the stationary points of List's flow are solutions to the static Einstein vacuum equations, and studied the extended parabolic system
\begin{align*}
\partial_{t}g &=-2\text{Ric}(g)+2\alpha_{n}d\varphi \otimes d\varphi , \\
\partial_{t}\varphi &=\Delta_{g}\varphi ,
\end{align*}
which is equivalent to the gradient flow of $E$. For applications on noncompact asymptotically flat manifolds, he proved short time existence on complete manifolds in the case when $\varphi $ is a smooth function from $M$ to $\mathbb{R}$.
In this paper the author studies List's flow on a compact manifold such that the scalar curvature is bounded. He establishes a time derivative bound for solutions to the heat equation, and derives the existence of a cutoff function (with good properties) whose time derivative and Laplacian are bounded. This can be seen as a parabolic version of Cheeger-Colding's cutoff function in the setting of Ricci lower bound.
Based on the above results, the author proves a backward pseudolocality theorem for the List's flow in dimension four. In this, the author needs to prove the $L^{\infty }$ estimate for subsolutions to nonhomogeneous linear heat equations along List's flow using Moser iteration method.
As an application, the author obtains that the $L^{2}$-norm of the Riemannian curvature operator is bounded and also gets the limit behavior of the List's flow. More precisely, based on an $L^{2}$-bound on the Riemannian curvature operator and a backward pseudolocality result, the author shows that if $M$ is a compact $4$-dimensional Riemannian manifold, $(M,g(t),\varphi (t)) $ is a List's flow on $[0,T)$, and if the trace of the Ricci tensor $S$ satisfies $|S|\leq 1$ on $M\times \lbrack 0,T),$ and $|\varphi_{0}|\leq 1$, then $(M,g(t),\varphi (t))$ converges to an orbifold in the Cheeger-Gromov sense as $t\rightarrow T$.
Reviewer: Boubaker-Khaled Sadallah (Algier)A study on conformal Ricci solitons and conformal Ricci almost solitons within the framework of almost contact geometryhttps://zbmath.org/1521.530772023-11-13T18:48:18.785376Z"Dey, S."https://zbmath.org/authors/?q=ai:dey.santu|dey.soumen|dey.sabyasachi|dey.subhasish|dey.subrata|dey.souvik|dey.sudip|dey.santanu-subhas|dey.smita|dey.sangeeta|dey.shubhamoy|dey.sandip|dey.sagar|dey.sujit|dey.subhrajit|dey.sandeep-kumar|dey.subhankar|dey.sumitra|dey.sourya|dey.subhrakanti|dey.siladitya|dey.saikat|dey.subir-kr|dey.satadru|dey.sourav|dey.sweta|dey.s-r|dey.satavisha|dey.sampa|dey.supravat|dey.shirsendu|dey.sushil-kumar|dey.soumyajit|dey.snigdhadip|dey.sanku|dey.sanjib|dey.shivani|dey.somajit|dey.santi-sekhar|dey.soumitra|dey.sudeep|dey.sukhendu|dey.shibshankar|dey.sanjana|dey.sitansu|dey.subhadip|dey.suhrit-k|dey.sumanta|dey.s-charlie|dey.samir|dey.sutirth\textit{A. Ghosh} [Carpathian Math. Publ. 11, No. 1, 59--69 (2019; Zbl 1423.53029)] studied Ricci solitons and Ricci almost solitons in the framework of Kenmotsu manifolds. In a similar way, the present author studies conformal Ricci solitons and conformal Ricci almost solitons in the framework of almost contact geometry.
The conformal Ricci flow is an analogue of the Navier-Stokes equation of fluid dynamics. Self similar solutions of the conformal Ricci flow are known as conformal Ricci solitons. Conformal Ricci almost solitons are generalizations of conformal Ricci solitons. The author proves that a Kenmotsu metric is Einstein if it is \(\eta\)-Einstein or the potential vector field is infinitesimal contact transformation. The results are illustrated by examples.
Reviewer: Uday Chand De (Kolkata)On the first eigenvalue of the Neumann problemhttps://zbmath.org/1521.580022023-11-13T18:48:18.785376Z"Meira, Adson"https://zbmath.org/authors/?q=ai:meira.adsonSummary: Let \(M^n\) be a smooth compact and connected Riemannian manifold with boundary \(\partial M \neq \emptyset\). Suppose \(M\) immersed into \(\mathbb{R}^{n+p}\), let \(\lambda_1 > 0\) be the first eigenvalue of the Neumann eigenvalue problem on \(M\), and \(H\) be the mean curvature of this immersion. Under a given condition, we prove that \(\lambda_1 vol(M) \leq n \int_M H^2 d V\), and the equality holds only if \(M\) is a minimal submanifold of some Euclidean hypersphere. This inequality was firstly proved by \textit{R. C. Reilly} [Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)]
for the closed eigenvalue problem.Non-degeneracy of critical points of the squared norm of the second fundamental form on manifolds with minimal boundaryhttps://zbmath.org/1521.580032023-11-13T18:48:18.785376Z"Cruz-Blázquez, Sergio"https://zbmath.org/authors/?q=ai:cruz-blazquez.sergio"Pistoia, Angela"https://zbmath.org/authors/?q=ai:pistoia.angelaAuthors' abstract: Let \((M, \overline{g})\) be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on \(\partial M\). We show that for a generic Riemannian metric \(\overline{g}\), the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate. We show that the generality of this property holds when we restrict ourselves to the conformal class of the initial metric on \(M\).
Reviewer: Marcelo Furtado (Brasília)Triharmonic CMC hypersurfaces in \({\mathbb{R}}^5(c)\)https://zbmath.org/1521.580052023-11-13T18:48:18.785376Z"Chen, Hang"https://zbmath.org/authors/?q=ai:chen.hang.1|chen.hang"Guan, Zhida"https://zbmath.org/authors/?q=ai:guan.zhidaSummary: A triharmonic map is a critical point of the tri-energy functional defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that any CMC proper triharmonic hypersurface in the 5-dimensional space form \({\mathbb{R}}^5(c)\) must have constant scalar curvature. Furthermore, we show that any CMC triharmonic hypersurface in \({\mathbb{R}}^5\) or \({\mathbb{H}}^5\) must be minimal, which supports the generalized Chen's conjecture; we also give some characterizations of CMC proper triharmonic hypersurfaces in \({\mathbb{S}}^5\). Similar results are obtained in the higher dimension case under an additional assumption on the numbers of the distinct principal curvatures.Diagrams and harmonic maps, revisitedhttps://zbmath.org/1521.580072023-11-13T18:48:18.785376Z"Pacheco, Rui"https://zbmath.org/authors/?q=ai:pacheco.rui"Wood, John C."https://zbmath.org/authors/?q=ai:wood.john-cIn the paper under review, the authors study and apply the finiteness criterion they have developed in collaboration with A. Aleman to extend many known results initially established for harmonic maps from the \(2\)-sphere into a Grassmannian
to harmonic maps from an arbitrary Riemann surface into a finite uniton number.
These results include: the description of harmonic maps of finite uniton number from any Riemann surface into a complex Grassmannian \(G_{2}(\mathbb{C}^{n})\). Noting that this extension relies on a new theory of nilpotent cycles. Also, a constancy result is developed showing that a harmonic map from a torus to a complex Grassmannian which is simultaneously of
finite uniton number and finite type is constant. Besides, authors generalize how harmonic maps of finite uniton number
can be constructed explicitly from holomorphic data from any Riemann surface instead of the \(2\)-sphere, using new methods.
Reviewer: Hiba Bibi (Tours)On the existence of solutions of a critical elliptic equation involving Hardy potential on compact Riemannian manifoldshttps://zbmath.org/1521.580082023-11-13T18:48:18.785376Z"Terki, Fatima Zohra"https://zbmath.org/authors/?q=ai:terki.fatima-zohra"Maliki, Youssef"https://zbmath.org/authors/?q=ai:maliki.youssefIn this paper, the authors study the existence of weak solutions to a certain non-linear elliptic equation on punctured compact Riemannian manifolds. This equation is a natural generalization of e.g. the equation for the scalar curvature in the Yamabe problem.
More precisely, let \((M,g)\) be an \(n\geq 3\) dimensional closed oriented Riemannian manifold with injectivity radius \(\delta_g>0\). Take a fixed point \(p\in M\) and define the truncated distance function \(\rho_p\) about this point to be the usual distance function in the geodesic ball \(B_{\delta_g}(p)\) and to be the constant \(\delta_g\) in the complementum \(M\setminus B_{\delta_g}(p)\). Let \(f,h\) be further smooth functions on \(M\) and consider the following nonlinear 2nd order scalar PDE on the punctured space \(M\setminus\{p\}\):
\[
\Delta_gu-\frac{h}{\rho_p^2}u=f\vert u\vert^{2^*-2}u
\]
where \(2^*=\frac{2n}{n-2}\) is the critical Sobolev exponent. Note that this equation gives back the Yamabe equation if the Hardy potential \(\frac{h}{\rho_p^2}\) is specified to \(\frac{n-2}{4(n-1)}\mathrm{Scal}_g\).
The authors study the existence of weak solutions to this equation over \(M\), more precisely solutions \(u\in L^2_1(M,g)\) or in different notation \(u\in H^2_1(M,g)\). For the precise technical statement cf. Theorems 1 and 2 in the paper.
Reviewer: Gábor Etesi (Budapest)On systoles and ortho spectrum rigidityhttps://zbmath.org/1521.580132023-11-13T18:48:18.785376Z"Masai, Hidetoshi"https://zbmath.org/authors/?q=ai:masai.hidetoshi"McShane, Greg"https://zbmath.org/authors/?q=ai:mcshane.gregA famous question asks if the spectrum of the Laplacian determines on a Riemann surface the geometry of the surface (``can one hear the shape of a drum?''). The answer to this question is no -- neither the length spectrum, nor the spectrum of the Laplacian determine the surface, but they do so up to finite ambiguity.
This article studies a question in the same circle of ideas. Namely, the authors consider a hyperbolic surface \(X\) with one geodesic boundary component, and ask to what extend the geometry is determined by the \textit{orthospectrum}, i.e. the lengths of all geodesics arcs meeting the boundary orthogonally.
It was known before that the length of the boundary, the area of the surface, and the entropy of the geodesic flow can be recovered [\textit{A. Basmajian}, Am. J. Math. 115, No. 5, 1139--1159 (1993; Zbl 0794.30032); \textit{M. Bridgeman}, Geom. Topol. 15, No. 2, 707--733 (2011; Zbl 1226.32007); \textit{S. P. Kerckhoff}, Ann. Math. (2) 117, 235--265 (1983; Zbl 0528.57008)]. The main theorems of this article show that the situation is the same as for the length spectrum: one can construct non-isometric surfaces with the same orthospectrum (Section 6), but there are only finitely many surfaces with the same orthospectrum (Theorem 5.1).
The examples in Section 6 use abelian covers to construct surfaces with the same orthospectrum, but different systoles. The proof of Theorem 5.1 follows a strategy by Wolpert, which proceeds by bounding the systole length from below using the orthospectrum. This is the part in which working with the orthospectrum is somewhat more subtle. The authors also include different arguments which only work for certain surfaces, which may nevertheless be of independent interest.
Reviewer: Sebastian Hensel (München)On intersection probabilities of four lines inside a planar convex domainhttps://zbmath.org/1521.600072023-11-13T18:48:18.785376Z"Martirosyan, Davit"https://zbmath.org/authors/?q=ai:martirosyan.davit-m"Ohanyan, Victor"https://zbmath.org/authors/?q=ai:ohanyan.v-kIn this paper, the authors consider the problem of computing the probabilities \(p_{nk}\) that \(n\) random lines produce exactly \(k\) intersection points inside a planar convex domain \(D\), and their approach is based on the use of Ambartzumian's combinatorial algorithm.
To illustrate this approach, the authors first re-derive existing results for \(n=2,3\). They then introduce new geometric invariants, which are subsequently used for the formulae for \(p_{4k}\), obtained by a careful study of events that lead to different number of intersections. The paper concludes with some explicit formulae in the case where \(D\) is a disc.
Reviewer: Mo Dick Wong (Durham)Covariant dynamics on the energy-momentum space: scalar field theoryhttps://zbmath.org/1521.811102023-11-13T18:48:18.785376Z"Ivetić, Boris"https://zbmath.org/authors/?q=ai:ivetic.borisSummary: A scalar field theory is constructed on an energy-momentum background of constant curvature. The generalization of the usual Feynamn rules for the flat geometry follows from the requirement of their covariance. The main result is that the invariant amplitudes are finite at all orders of the perturbation theory, due to the finiteness of the momentum space. Finally, the relation with a field theory in spacetime representation is briefly discussed.Curvature-dimension conditions for symmetric quantum Markov semigroupshttps://zbmath.org/1521.811192023-11-13T18:48:18.785376Z"Wirth, Melchior"https://zbmath.org/authors/?q=ai:wirth.melchior"Zhang, Haonan"https://zbmath.org/authors/?q=ai:zhang.haonanSummary: Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras and generalized depolarizing semigroups.Cyclic cocycles and one-loop corrections in the spectral actionhttps://zbmath.org/1521.811232023-11-13T18:48:18.785376Z"van Nuland, Teun D. H."https://zbmath.org/authors/?q=ai:van-nuland.teun-d-h"van Suijlekom, Walter D."https://zbmath.org/authors/?q=ai:van-suijlekom.walter-danielSummary: We present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd \((b,B)\)-cocycle, whereas, in the even orders, powers of the curvature are integrated against \((b,B)\)-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action \(\mathrm{Tr}(f(D+V))\) in powers of \(V=\pi_D(A)\), but unlike the Taylor expansion we expand in increasing order of the forms in \(A\). We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.
For the entire collection see [Zbl 1507.19001].The full decision of Yang-Mills equations for the central-symmetric metricshttps://zbmath.org/1521.811742023-11-13T18:48:18.785376Z"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: This article develops the researches begun in [the authors, Russ. Math. 53, No. 9, 62--66 (2009; Zbl 1183.81106); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 9, 69--74 (2009)]. The system of the differential Yang-Mills equations for the central-symmetric metrics is deduced. The general solution of this system expressed through elliptic \(\wp \)-function of Weierstrass is resulted. For several special cases the solutions expressed through elementary functions are received. Criteria that the metrics which is the direct sum of two binary quadratic forms is Einstein metrics and the conformally-flat metrics are proved.\( \text{Spin}^c\)-structures and Seiberg-Witten equationshttps://zbmath.org/1521.811752023-11-13T18:48:18.785376Z"Sergeev, A. G."https://zbmath.org/authors/?q=ai:sergeev.armen-glebovichSummary: The Seiberg-Witten equations, found at the end of the 20th century, are one of the main discoveries in the topology and geometry of four-dimensional Riemannian manifolds. They are defined in terms of a \(\text{Spin}^c\)-structure that exists on any four-dimensional Riemannian manifold. Like the Yang-Mills equations, the Seiberg-Witten equations are the limit case of a more general supersymmetric Yang-Mills equations. However, unlike the conformally invariant Yang-Mills equations, the Seiberg-Witten equations are not scale invariant. Therefore, in order to obtain ``useful information'' from them, one must introduce a scale parameter \(\lambda\) and pass to the limit as \(\lambda\to\infty \). This is precisely the adiabatic limit studied in this paper.Symplectic modular symmetry in heterotic string vacua: flavor, CP, and \(R\)-symmetrieshttps://zbmath.org/1521.812312023-11-13T18:48:18.785376Z"Ishiguro, Keiya"https://zbmath.org/authors/?q=ai:ishiguro.keiya"Kobayashi, Tatsuo"https://zbmath.org/authors/?q=ai:kobayashi.tatsuo"Otsuka, Hajime"https://zbmath.org/authors/?q=ai:otsuka.hajimeSummary: We examine a common origin of four-dimensional flavor, CP, and \(\mathrm{U}(1)_{\mathrm{R}}\) symmetries in the context of heterotic string theory with standard embedding. We find that flavor and \(\mathrm{U}(1)_{\mathrm{R}}\) symmetries are unified into the \(\mathrm{Sp}(2h + 2, \mathbb{C})\) modular symmetries of Calabi-Yau threefolds with \(h\) being the number of moduli fields. Together with the \(\mathbb{Z}_2^{\mathrm{CP}}\) CP symmetry, they are enhanced to \(G\mathrm{Sp}(2h + 2, \mathbb{C}) \simeq\mathrm{Sp}(2h + 2, \mathbb{C})\rtimes\mathbb{Z}_2^{\mathrm{CP}}\) generalized symplectic modular symmetry. We exemplify the \(S_3\), \(S_4\), \(T^\prime\), \(S_9\) non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and \(\mathbb{Z}_2\), \(S_4\) flavor symmetries on three-parameter examples of Calabi-Yau threefolds. Thus, non-trivial flavor symmetries appear in not only the exact orbifold limit but also a certain class of Calabi-Yau three-folds. These flavor symmetries are further enlarged to non-Abelian discrete groups by the CP symmetry.Note on the bundle geometry of field space, variational connections, the dressing field method, \& presymplectic structures of gauge theories over bounded regionshttps://zbmath.org/1521.813412023-11-13T18:48:18.785376Z"François, J."https://zbmath.org/authors/?q=ai:francois.jordan"Parrini, N."https://zbmath.org/authors/?q=ai:parrini.n"Boulanger, N."https://zbmath.org/authors/?q=ai:boulanger.nicolasSummary: In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the ``dressing field method''. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.A hyperbolic analogue of the Atiyah-Hitchin manifoldhttps://zbmath.org/1521.813512023-11-13T18:48:18.785376Z"Sutcliffe, Paul"https://zbmath.org/authors/?q=ai:sutcliffe.paul-m|sutcliffe.paul-jSummary: The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric charge two SU(2) monopoles in Euclidean space. Here a hyperbolic analogue is presented, by calculating the boundary metric on the moduli space of parity inversion symmetric charge two SU(2) monopoles in hyperbolic space. The calculation of the metric is performed using a twistor description of the moduli space and the result is presented in terms of standard elliptic integrals.General relativity from \(p\)-adic stringshttps://zbmath.org/1521.830062023-11-13T18:48:18.785376Z"Huang, An"https://zbmath.org/authors/?q=ai:huang.an"Stoica, Bogdan"https://zbmath.org/authors/?q=ai:stoica.bogdan"Yau, Shing-Tung"https://zbmath.org/authors/?q=ai:yau.shing-tungSummary: For an arbitrary prime number \(p\), we propose an action for bosonic \(p\)-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum \(p\)-adic strings, similar to the ordinary bosonic strings case. It turns out that spherical vectors of unramified principal series representations of \(PGL(2,\mathbb{Q}_p)\) are the plane wave modes of the bosonic fields on \(p\)-adic strings, and that the regularized normalization of these modes on the padic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.Weyl curvature evolution system for GRhttps://zbmath.org/1521.830332023-11-13T18:48:18.785376Z"Krasnov, Kirill"https://zbmath.org/authors/?q=ai:krasnov.kirill-v"Shaw, Adam"https://zbmath.org/authors/?q=ai:shaw.adamSummary: Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree \(3 \times 3\) matrix \(\Psi^{ij}\), which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial \(\mathrm{SO}(3, \mathbb{C})\) connection \(A^i_a\). The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field \(\Psi^{ij}\) and the curvature of the spatial connection \(A^i_a\). The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are the most natural spin two generalisations of Maxwell's spin one equations. We also determine the modifications of the evolution system needed to enforce the `constraint sweeping', so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.The gravito-electromagnetic approximation to the gravimagnetic dipole and its velocity rotation curvehttps://zbmath.org/1521.830702023-11-13T18:48:18.785376Z"Govaerts, Jan"https://zbmath.org/authors/?q=ai:govaerts.janSummary: In view of the observed flat rotation curves of spiral galaxies and motivated by the simple fact that within Newtonian gravity a stationary axisymmetric mass distribution or dark matter vortex of finite extent readily displays a somewhat flattened out velocity rotation curve up to distances comparable to the extent of such a vortex transverse to the galaxy's disk, the possibility that such a flattening out of rotation curves may rather be a manifestation of some stationary axisymmetric space-time curvature of purely gravitational character, without the need of some dark matter particles, is considered in the case of the gravimagnetic dipole carrying opposite
Newman-Unti-Tamburino charges and in the tensionless limit of its Misner string, as an exact vacuum solution to Einstein's equations. Aiming for a first assessment of the potential of such a suggestion easier than a full-fledged study of its geodesics, the situation is analysed within the limits of weak field gravito-electromagnetism and nonrelativistic dynamics. Thereby leading indeed to interesting and encouraging results.Special Vinberg cones and the entropy of BPS extremal black holeshttps://zbmath.org/1521.830792023-11-13T18:48:18.785376Z"Alekseevsky, Dmitri V."https://zbmath.org/authors/?q=ai:alekseevskii.dmitri-vladimirovich"Marrani, Alessio"https://zbmath.org/authors/?q=ai:marrani.alessio"Spiro, Andrea"https://zbmath.org/authors/?q=ai:spiro.andrea-fSummary: We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged \(N = 2\) \(D = 4\) supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg's theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of \(N = 2\) supergravity with homogeneous scalar manifold.Scale-invariance at the core of quantum black holeshttps://zbmath.org/1521.830932023-11-13T18:48:18.785376Z"Borissova, Johanna N."https://zbmath.org/authors/?q=ai:borissova.johanna-n"Held, Aaron"https://zbmath.org/authors/?q=ai:held.aaron"Afshordi, Niayesh"https://zbmath.org/authors/?q=ai:afshordi.niayeshSummary: We study spherically-symmetric solutions to a modified Einstein-Hilbert action with renormalization group (RG) scale-dependent couplings, inspired by Weinberg's Asymptotic Safety scenario for quantum gravity. The RG scale is identified with the Tolman temperature for an isolated gravitational system in thermal equilibrium with Hawking radiation. As a result, the point of infinite local temperature is shifted from the classical black-hole horizon to the origin and coincides with a timelike curvature singularity. Close to the origin, the spacetime is determined by the scale-dependence of the cosmological constant in the vicinity of the Reuter fixed point: the free components of the metric can be derived analytically and are characterized by a radial power law with exponent \(\alpha = \sqrt{3} - 1\). Away from the fixed point, solutions for different masses are studied numerically and smoothly interpolate between the Schwarzschild exterior and the scale-invariant interior. Whereas the exterior of objects with astrophysical mass is described well by vacuum general relativity, deviations become significant at a Planck distance away from the classical horizon and could lead to observational signatures. We further highlight potential caveats in this intriguing result with regard to our choice of scale-identification and identify future avenues to better understand quantum black holes in relation to the key feature of scale-invariance.Regular black holes with sub-Planckian curvaturehttps://zbmath.org/1521.831312023-11-13T18:48:18.785376Z"Ling, Yi"https://zbmath.org/authors/?q=ai:ling.yi"Wu, Meng-He"https://zbmath.org/authors/?q=ai:wu.meng-heSummary: We construct a sort of regular black holes with a sub-Planckian Kretschmann scalar curvature. The metric of this sort of regular black holes is characterized by an exponentially suppressing gravity potential as well as an asymptotically Minkowski core. In particular, with different choices of the potential form, they can reproduce the metric of Bardeen/Hayward/Frolov black hole at large scales. The heuristical derivation of this sort of black holes is performed based on the generalized uncertainty principle over curved spacetime which includes the effects of tidal force on any object with finite size which is bounded below by the minimal length.Vacuum-dual static perfect fluid obeying \(p = -(n-3)\rho/(n+1)\) in \(n(\geqslant 4)\) dimensionshttps://zbmath.org/1521.831372023-11-13T18:48:18.785376Z"Maeda, Hideki"https://zbmath.org/authors/?q=ai:maeda.hidekiSummary: We obtain the general \(n(\geqslant 4)\)-dimensional static solution with an \(n-2\)-dimensional Einstein base manifold for a perfect fluid obeying a linear equation of state \(p = -(n-3)\rho/(n+1)\). It is a generalization of Semiz's four-dimensional general solution with spherical symmetry and consists of two different classes. Through the Buchdahl transformation, the class-I and class-II solutions are dual to the topological Schwarzschild-Tangherlini-(A)dS solution and one of the \(\Lambda\)-vacuum direct-product solutions, respectively. While the metric of the spherically symmetric class-I solution is \(C^\infty\) at the Killing horizon for \(n = 4\) and 5, it is \(C^1\) for \(n \geqslant 6\) and then the Killing horizon turns to be a parallelly propagated curvature singularity. For \(n = 4\) and 5, the spherically symmetric class-I solution can be attached to the Schwarzschild-Tangherlini vacuum black hole with the same value of the mass parameter at the Killing horizon in a regular manner, namely without a lightlike massive thin-shell. This construction allows new configurations of an asymptotically (locally) flat black hole to emerge. If a static perfect fluid hovers outside a vacuum black hole, its energy density is negative. In contrast, if the dynamical region inside the event horizon of a vacuum black hole is replaced by the class-I solution, the corresponding matter field is an anisotropic fluid and may satisfy the null and strong energy conditions. While the latter configuration always involves a spacelike singularity inside the horizon for \(n = 4\), it becomes a non-singular black hole of the big-bounce type for \(n = 5\) if the ADM mass is larger than a critical value.Adaptive polygon rendering for interactive visualization in the Schwarzschild spacetimehttps://zbmath.org/1521.831432023-11-13T18:48:18.785376Z"Müller, Thomas"https://zbmath.org/authors/?q=ai:muller.thomas.5"Schulz, Christoph"https://zbmath.org/authors/?q=ai:schulz.christoph.2"Weiskopf, Daniel"https://zbmath.org/authors/?q=ai:weiskopf.daniel(no abstract)BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversionhttps://zbmath.org/1521.831532023-11-13T18:48:18.785376Z"van Geemen, Bert"https://zbmath.org/authors/?q=ai:van-geemen.bert"Marrani, Alessio"https://zbmath.org/authors/?q=ai:marrani.alessio"Russo, Francesco"https://zbmath.org/authors/?q=ai:russo.francesco.1Summary: We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of \(\mathcal{N} = 2\), \(D = 4\) ungauged supergravity obtained as reduction of minimal, matter-coupled \(D = 5\) supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named \textit{BPS system}, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous \textit{non-symmetric} scalar manifolds (whose classification is known in terms of \(L(q, P, \dot{P})\) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an \textit{invertibility condition} for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with \(\dot{P} = 0\), such as \(L(q, P)\), \(1\leqslant q \leqslant 3\) and \(P \geqslant 1\), or \(L(q, 1)\), \(4 \leqslant q \leqslant 9\), and one model with \(\dot{P} = 1\), namely \(L(4, 1, 1)\). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of \(L(1, P)\) \(P\geqslant2\) non-symmetric models of \(\mathcal{N} = 2\), \(D = 4\) supergravity.Geometrical effective actions for a partially massless spin-2 fieldhttps://zbmath.org/1521.831612023-11-13T18:48:18.785376Z"Kan, Nahomi"https://zbmath.org/authors/?q=ai:kan.nahomi"Aoyama, Takuma"https://zbmath.org/authors/?q=ai:aoyama.takuma"Shiraishi, Kiyoshi"https://zbmath.org/authors/?q=ai:shiraishi.kiyoshiSummary: We consider nonlinear effective actions for a spin-2 field, whose `decoupling' limit gives Fierz-Pauli action in \(D\) dimensional maximally symmetric spacetime. We find, especially, the effective action for a partially massless field can take a concise geometrical form. The exact solution for time evolution of the background metric in the model using the effective action is also studied.Kähler-Einstein metrics near an isolated log-canonical singularityhttps://zbmath.org/1521.831642023-11-13T18:48:18.785376Z"Datar, Ved"https://zbmath.org/authors/?q=ai:datar.ved-v"Fu, Xin"https://zbmath.org/authors/?q=ai:fu.xin.1"Song, Jian"https://zbmath.org/authors/?q=ai:song.jianSummary: We construct Kähler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler-Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler-Einstein metrics from hyperbolic geometry.On pseudoconvexity conditions and static spacetimeshttps://zbmath.org/1521.832182023-11-13T18:48:18.785376Z"Vatandoost, Mehdi"https://zbmath.org/authors/?q=ai:vatandoost.mehdi"Pourkhandani, Rahimeh"https://zbmath.org/authors/?q=ai:pourkhandani.rahimeh(no abstract)Strange quark star models from Rastall gravityhttps://zbmath.org/1521.850012023-11-13T18:48:18.785376Z"Banerjee, Ayan"https://zbmath.org/authors/?q=ai:banerjee.ayan"Tangphati, Takol"https://zbmath.org/authors/?q=ai:tangphati.takol"Hansraj, Sudan"https://zbmath.org/authors/?q=ai:hansraj.sudan"Pradhan, Anirudh"https://zbmath.org/authors/?q=ai:pradhan.anirudhSummary: We study the possible existence of strange matter in the interior of compact stars in Rastall gravity. The main feature of the Rastall gravity is the covariant derivative of the energy-momentum tensor does not vanish and depends on the curvature \(R\) and a free parameter \(\eta\). We consider the MIT bag model in the present work and numerically solve the modified Tolman-Oppenheimer-Volkoff (TOV) equations. We investigate the effect of the Rastall parameter \(\eta\) on the mass-radius and the mass-central energy density relation of quark stars. From the quark star masses and radii obtained, we conclude that \(\eta\) has significant consequences on the structure of stellar objects.Small-scale dynamo in Riemannian spaces of constant curvaturehttps://zbmath.org/1521.860452023-11-13T18:48:18.785376Z"Sokoloff, Dmitry"https://zbmath.org/authors/?q=ai:sokolev.dmitry|sokolov.dmitrii-dmitrievich"Rubashny, Alexey"https://zbmath.org/authors/?q=ai:rubashny.alexeySummary: We compare temporal growth of the mean magnetic energy \(\mathcal{E}\) driven by a small-scale dynamo in Euclidean and Lobachevsky spaces. The governing parameters of the dynamo, such as the rms turbulent velocity and the correlation scale of turbulence, are presumed to vary randomly in space so the dynamo growth rate is a Gaussian random field. Since such a field is unbounded in unbounded space and can achieve, with a low probability, very large values of \(\mathcal{E}\), it can grow super-exponentially in both cases. The super-exponential growth of \(\mathcal{E}\) in Euclidean space, known since the 1980s, can be considered as a statement that the mean energy growth rate is determined up to a weakly growing factor proportional to \(\sqrt{\ln t}\). We demonstrate that the super-exponential growth of \(\mathcal{E}\) in Lobachevsky space is a much more radical phenomenon, where \(\mathcal{E}\) grows as \(\exp(\mathrm{const}\times t^{5/3})\). We stress that extrapolating the properties of small-scale dynamos in Euclidean space to curved geometries such Lobachevsky space is not straightforward and requires some care. The effects under discussion becomes however important only if the spatial scale of domain in which the small-scale magnetic field is excited exceeds the radius of curvature.