Recent zbMATH articles in MSC 53Chttps://zbmath.org/atom/cc/53C2022-09-13T20:28:31.338867ZUnknown authorWerkzeugBook review of: J. D. Moore, Introduction to global analysis. Minimal surfaces in Riemannian manifoldshttps://zbmath.org/1491.000092022-09-13T20:28:31.338867Z"Colding, Tobias Holck"https://zbmath.org/authors/?q=ai:colding.tobias-holckReview of [Zbl 1484.53002].Delta invariants of projective bundles and projective cones of Fano typehttps://zbmath.org/1491.140622022-09-13T20:28:31.338867Z"Zhang, Kewei"https://zbmath.org/authors/?q=ai:zhang.kewei"Zhou, Chuyu"https://zbmath.org/authors/?q=ai:zhou.chuyuThe \(\delta\)-invariant (also known as the stability threshold) of a Fano variety characterizes its K-stability. It was proved by \textit{K. Fujita} and \textit{Y. Odaka} [Tohoku Math. J. (2) 70, No. 4, 511--521 (2018; Zbl 1422.14047)] and \textit{H. Blum} and \textit{M. Jonsson} [Adv. Math. 365, Article ID 107062, 57 p. (2020; Zbl 1441.14137)] that a Fano variety \(X\) is K-semistable (resp. uniformly K-stable) if and only if \(\delta(X) > 1\) (resp. \(\delta(X) \geq 1\)). If \(X\) is not uniformly K-stable, the \(\delta\)-invariant equals the greatest Ricci lower bound as shown in [\textit{I. A. Cheltsov} et al., Sel. Math., New Ser. 25, No. 2, Paper No. 34, 36 p. (2019; Zbl 1418.32015); \textit{R. J. Berman} et al., J. Am. Math. Soc. 34, No. 3, 605--652 (2021; Zbl 1487.32141)]. In this article, the authors compute \(\delta\)-invariant of \(\mathbb{P}^1\)-bundles and projective cones over a Fano variety \(V\).
Suppose \(V\) is a Fano variety of dimension \(n\) and Fano index \(\geq 2\). Let \(L = -\frac{1}{r}K_V\) be an ample line bundle for some rational number \(r>1\). Let \(\tilde{Y}:=\mathbb{P}_V(L^{-1} \oplus\mathcal{O}_V)\) be the \(\mathbb{P}^1\)-bundle as the compactification of the total space of \(L^{-1}\). Let
\[
\beta_0:= \left(\frac{n+1}{n+2}\cdot \frac{(r+1)^{n+2} - (r-1)^{n+2}}{(r+1)^{n+1} - (r-1)^{n+1}} - (r-1)\right)^{-1}.
\]
Theorem 1.1 states that
\[
\delta(\tilde{Y}) = \min \left\{\frac{\delta(V)r\beta_0}{1+\beta_0(r-1)}, \beta_0\right\}.
\]
Let \(Y\) be the projective cone over \(V\) with polarization \(L\). Then \(\tilde{Y}\to Y\) is a birational morphism that contracts the zero section \(V_0\) to the cone point in \(Y\). Theorem 1.4 implies that
\[
\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}\min\left\{1, \delta(V), \frac{n+1}{r}\right\}.
\]
In particular, if \(V\) is K-semistable then \(r\leq n+1\) by \textit{K. Fujita} [Am. J. Math. 140, No. 2, 391--414 (2018; Zbl 1400.14105)], so we have \(\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}\). Similar computations of \(\delta\)-invariants are also done for log Fano pairs \((\tilde{Y}, aV_0+bV_\infty)\) and \((Y, cV_\infty)\). Applications are included for computations of \(\delta\)-invariants of certain singular hypersurfaces, and the existence of conical Kähler-Einstein metrics on projective spaces with certain cone angle along a smooth Fano hypersurface.
Reviewer: Yuchen Liu (Evanston)The Kapustin-Witten equations and nonabelian Hodge theoryhttps://zbmath.org/1491.140642022-09-13T20:28:31.338867Z"Liu, Chih-Chung"https://zbmath.org/authors/?q=ai:liu.chih-chung"Rayan, Steven"https://zbmath.org/authors/?q=ai:rayan.steven"Tanaka, Yuuji"https://zbmath.org/authors/?q=ai:tanaka.yuujiSummary: Arising from a topological twist of \({\mathscr{N}}=4\) super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrised by \(t\in{\mathbb{P}}^1\). The parameter corresponds to a linear combination of two super charges in the twist. When \(t=0\) and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of \(\lambda \)-connection in the nonabelian Hodge theory of Donaldson-Corlette-Hitchin-Simpson in which \(\lambda\) is also valued in \({\mathbb{P}}^1\). Varying \(\lambda\) interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at \(\lambda =0)\) and the moduli space of semisimple local systems on the same variety (at \(\lambda =1)\) in the twistor space. In this article, we utilise the correspondence furnished by nonabelian Hodge theory to describe a relation between the moduli spaces of solutions to the equations by Kapustin and Witten at \(t=0\) and \(t \in{{\mathbb{R}}} \,{\setminus}\, \{ 0 \}\) on a smooth, compact Kähler surface. We then provide supporting evidence for a more general form of this relation on a smooth, closed four-manifold by computing its expected dimension of the moduli space for each of \(t=0\) and \(t \in{{\mathbb{R}}} \,{\setminus}\, \{ 0 \} \).A quantum Leray-Hirsch theorem for banded gerbeshttps://zbmath.org/1491.140792022-09-13T20:28:31.338867Z"Tang, Xiang"https://zbmath.org/authors/?q=ai:tang.xiang"Tseng, Hsian-Hua"https://zbmath.org/authors/?q=ai:tseng.hsian-huaThe relationship between global and local is an eternal theme of mathematics. The classical Künneth formula expresses the cohomology of a product space as a tensor product of the cohomologies of the direct factors. Furthermore, if \(\pi: E\rightarrow B\) is a fiber bundle with fiber \(F\), the celebrated Leray-Hirsch theorem states that the cohomology of fiber bundle \(E\) is equal to the tensor product of the cohomologies of the base and fiber, i.e. \(H^* (E)\cong H^*(B)\otimes H^*(F)\). With the development of Gromov-Witten theory, a natural question is to ask : What is quantum geometry picture for the Leray-Hirsch theorem?
The paper under review tries to answer this question and studies the quantum version of Leray-Hirsch theorem in the context of orbifold Gromov-Witten theory of a gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) banded by a finite group \(G\), where \(\mathcal{B}\) is a smooth proper (i.e. compact) Deligne-Mumford stack and a \(G\)-gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) can be regarded as a fiber bundle over complex orbifold \(\mathcal{B}\) with fiber \(BG\). The main result (i.e. quantum Leray-Hirsch theorem) states that the Gromov-Witten theory of \(\mathcal{Y}\) can be expressed in terms of the Gromov-Witten theory of base \(\mathcal{B}\) and the information of fiber \(BG\). The key point of the proof is to analyze the properties and degree for the pushforward map \(\pi: \overline{\mathcal{M}}_{g, n}(\mathcal{Y}, \beta)\rightarrow \overline{\mathcal{M}}_{g, n}(\mathcal{B}, \beta)\).
In conclusion, the paper under review presents an important structural decomposition theorem for a banded \(G\)-gerbe over complex orbifold \(\mathcal{B}\) which can be regarded as a quantum Leray-Hirsch style theorem. The combinatorial degree formula and virtual pushforward formula are impressive in the proof.
Reviewer: Xiaobin Li (Chengdu)Smooth loops and loop bundleshttps://zbmath.org/1491.201552022-09-13T20:28:31.338867Z"Grigorian, Sergey"https://zbmath.org/authors/?q=ai:grigorian.sergeySummary: A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Maurer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also develop aspects of a non-associative gauge theory. Throughout, we see how some of the known properties of \(G_2\)-structures can be seen from this more general setting.Conformal symmetry breaking differential operators on differential formshttps://zbmath.org/1491.220042022-09-13T20:28:31.338867Z"Fischmann, Matthias"https://zbmath.org/authors/?q=ai:fischmann.matthias"Juhl, Andreas"https://zbmath.org/authors/?q=ai:juhl.andreas"Somberg, Petr"https://zbmath.org/authors/?q=ai:somberg.petrIn [\textit{A. Juhl}, Families of conformally covariant differential operators, Q-curvature and holography. Basel: Birkhäuser (2009; Zbl 1177.53001)], a program was initiated to find conformal invariants of hypersurfaces \(M\) in Riemannian manifolds \((X,g)\). The idea is to construct one-parameter families of conformally covariant differential operators \(D_{2N}(g;\lambda):C^\infty(X)\to C^\infty(M)\), \(\lambda\in\mathbb{C}\), of order \(2N\), mapping functions on the ambient manifold \(X\) to functions on \(M\). Here conformal covariance means that \[ e^{(\lambda+N)\iota^*(\varphi)}D_{2N}(e^{2\varphi}g;\lambda)(u) = D_{2N}(g;\lambda)(e^{\lambda\varphi}u) \] for all \(u,\varphi\in C^\infty(X)\), where \(\iota^*\) denotes the pull-back defined by the embedding \(\iota:M\hookrightarrow X\). Such families generalize the even-order families \(D_{2N}(\lambda):C^\infty(S^n)\to C^\infty(S^{n-1})\) of differential operators which are associated to the equatorial embedding \(S^{n-1}\hookrightarrow S^n\) of spheres, and which intertwine spherical principal series representations of the conformal group of the embedded round sphere \(S^{n-1}\), but not of the conformal group of the ambient round sphere \(S^n\). These intertwining families were first constructed in [\textit{A. Juhl}, Families of conformally covariant differential operators, Q-curvature and holography. Basel: Birkhäuser (2009; Zbl 1177.53001)] and can be regarded as differential operators \[ D_{2N}(\lambda):C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^{n-1}) \] via the stereographic projection.\\
In the paper under review, the authors consider the corresponding problem for differential operators between differential forms. They completely classify conformally covariant differential operators \[ \Omega^p(\mathbb{R}^n)\to\Omega^q(\mathbb{R}^{n-1}) \] for all degrees \(p\) and \(q\). These are intertwining operators for the conformal group of \(\mathbb{R}^{n-1}\), acting by principal series representations realized on forms, and correspond to homomorphisms of generalized Verma modules of the corresponding Lie algebra \(\mathfrak{so}(n,1)\) into Verma modules of \(\mathfrak{so}(n+1,1)\).
The classification essentially consists of two families of differential operators \[ D_N^{(p\to p)}(\lambda):\Omega^p(\mathbb{R}^n)\to\Omega^p(\mathbb{R}^{n-1}) \quad \mbox{and} \quad D_N^{(p\to p-1)}(\lambda):\Omega^p(\mathbb{R}^n)\to\Omega^{p-1}(\mathbb{R}^{n-1}), \] of degree \(N\in\mathbb{N}\), both depending on a complex parameter \(\lambda\in\mathbb{C}\), some additional sporadic operators \(\Omega^p(\mathbb{R}^n)\to\Omega^q(\mathbb{R}^{n-1})\) for \(q\in\{p-2,p-1,p,p+1\}\), as well as their compositions with the Hodge star operator \(\star\) on \(\mathbb{R}^n\) and \(\mathbb{R}^{n-1}\). The authors further obtain some relations between compositions of these operators with the exterior derivative \(d\) and its adjoint \(\delta\) on \(\mathbb{R}^n\) and \(\mathbb{R}^{n-1}\) which they refer to as factorization identities. Finally, they relate their operators to the critical \(Q\)-curvature operators and the Gauge companion operators.\\
The key technique used in the classification if the so-called F-method introduced in [\textit{T. Kobayashi}, Contemp. Math. 598, 139--146 (2013; Zbl 1290.22008) ] which characterizes the symbols of conformally covariant differential operators as (possibly vector-valued) polynomial solutions to certain differential equations. In the case at hand, the corresponding differential equations can be transformed into ordinary differential equations, explaining the occurrence of Jacobi polynomials in the explicit formulas for the two families \(D_N^{(p\to p)}(\lambda)\) and \(D_N^{(p\to p-1)}(\lambda)\).\\
We remark that the same classification was independently obtained in [\textit{T. Kobayashi} et al., Conformal symmetry breaking operators for differential forms on spheres. Singapore: Springer (2016; Zbl 1353.53002)].
Reviewer: Jan Frahm (Århus)Hamiltonian chaos and differential geometry of configuration space-timehttps://zbmath.org/1491.370532022-09-13T20:28:31.338867Z"Di Cairano, Loris"https://zbmath.org/authors/?q=ai:di-cairano.loris"Gori, Matteo"https://zbmath.org/authors/?q=ai:gori.matteo"Pettini, Giulio"https://zbmath.org/authors/?q=ai:pettini.giulio"Pettini, Marco"https://zbmath.org/authors/?q=ai:pettini.marcoSummary: This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space-time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi-Levi-Civita (JLC) equation for geodesic spread. It is confirmed that the dominant mechanism at the ground of chaotic dynamics is parametric instability due to curvature variations along the geodesics. A comparison is reported of the outcomes of the JLC equation written also for the Jacobi metric on configuration space and for another metric due to Eisenhart on an extended configuration space-time. This has been applied to the Hénon-Heiles model, a two-degrees of freedom system. Then the study has been extended to the 1D classical Heisenberg \(X Y\) model at a large number of degrees of freedom. Both the advantages and drawbacks of this geometrization of Hamiltonian dynamics are discussed. Finally, a quick hint is put forward concerning the possible extension of the differential-geometric investigation of chaos in generic dynamical systems, including dissipative ones, by resorting to Finsler manifolds.Aperiodic order and spherical diffraction. II: Translation bounded measures on homogeneous spaceshttps://zbmath.org/1491.430072022-09-13T20:28:31.338867Z"Björklund, Michael"https://zbmath.org/authors/?q=ai:bjorklund.michael"Hartnick, Tobias"https://zbmath.org/authors/?q=ai:hartnick.tobias"Pogorzelski, Felix"https://zbmath.org/authors/?q=ai:pogorzelski.felixSummary: We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on \(\mathbb{R}^n\). This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.
For Part I and III, see [the authors, Proc. Lond. Math. Soc. (3) 116, No. 4, 957--996 (2018; Zbl 1394.52022); J. Funct. Anal. 281, No. 12, Article ID 109265, 59 p. (2021; Zbl 1485.43005)].On a theorem of Chernoff on rank one Riemannian symmetric spaceshttps://zbmath.org/1491.430082022-09-13T20:28:31.338867Z"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Manna, Ramesh"https://zbmath.org/authors/?q=ai:manna.ramesh"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramSummary: In 1975, P. R. Chernoff used iterates of the Laplacian on \(\mathbb{R}^n\) to prove an \(L^2\) version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on \(\mathbb{R}^n\) to be quasi-analytic. In this paper we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces of noncompact type using iterates of the associated Laplace-Beltrami operators. Moreover, we also prove an analogue of Chernoff's theorem for the sphere which is a rank one compact symmetric space.Lower bound of sectional curvature of Fisher-Rao manifold of beta distributions and complete monotonicity of functions involving polygamma functionshttps://zbmath.org/1491.440022022-09-13T20:28:31.338867Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher-Rao manifold of beta distributions.Area minimizing surfaces in homotopy classes in metric spaceshttps://zbmath.org/1491.490282022-09-13T20:28:31.338867Z"Soultanis, Elefterios"https://zbmath.org/authors/?q=ai:soultanis.elefterios"Wenger, Stefan"https://zbmath.org/authors/?q=ai:wenger.stefanSummary: We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class.
Our theorems generalize and strengthen results of \textit{L. Lemaire} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 9, 91--104 (1982; Zbl 0532.58004)], \textit{J. Jost} [J. Reine Angew. Math. 359, 37--54 (1985; Zbl 0568.49025)], \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math. (2) 110, 127--142 (1979; Zbl 0431.53051)], and \textit{J. Sacks} and \textit{K. Uhlenbeck} [Trans. Am. Math. Soc. 271, 639--652 (1982; Zbl 0527.58008)].Canal surfaces and foliations -- a surveyhttps://zbmath.org/1491.530012022-09-13T20:28:31.338867Z"Walczak, Pawel"https://zbmath.org/authors/?q=ai:walczak.pawel-grzegorzIn this survey the author recollects several results (stated without proofs) on canal foliations, i.e., (codimension-1) foliations on 3-manifolds of constant curvature, whose (2-dimensional) leaves are canal surfaces. In turn, canal surfaces are envelopes of a family of spheres whose centres lie on a curve in \(\mathbb{R}^3\); examples include the torus, the cylinder, and other surfaces of revolution, as well as their images under Möbius transformations.
In the first two sections the author recalls how to interpret oriented 2-spheres in \(\mathbb{S}^3\) as points in the 4-dimensional De Sitter space \(\Delta^4 \subseteq \mathbb{R}^5\) (Section 2) and reviews some facts on the conformal geometry of surfaces (Section 3). This allows him to recast canal surfaces as space-like curves in \(\Delta^4\) and to characterise a few significant classes of examples (Section 4).
In Section 5 foliations enters into the picture. The first main theorem (proved in [\textit{R. Langevin} and \textit{P. G. Walczak}, J. Math. Soc. Japan 64, No. 2, 659--682 (2012; Zbl 1297.53021)]) shows that the classical Reeb foliation plays a fundamental role in the classification of canal foliations on \(\mathbb{S}^3\). Indeed, any such foliation is either the Reeb foliation itself or is obtained from it via a turbulisation procedure.
The existence problem for canal foliations on other 3-manifolds is less straightforward, therefore one restricts to particular cases. Another result (proved in [\textit{R. Langevin} and \textit{P. G. Walczak}, Geom. Dedicata 132, 135--178 (2008; Zbl 1143.53028)]) concerns the existence of Dupin foliations, i.e., foliations by Dupin cyclides (a simple but interesting class of canal surfaces). Such foliations never exist on closed hyperbolic manifolds, and they are trivial on \(\mathbb{R}^3\).
The third main theorem (proved in [\textit{G. Hector} et al., J. Math. Soc. Japan 71, No. 1, 43--63 (2019; Zbl 1417.57024)]) provides the complete list of all compact 3-manifolds admitting a \textit{topological} canal foliation (a generalisation of ``geometric'' canal foliations, which captures their underlying topological properties). As a corollary, since no manifold of such list admits a hyperbolic structure, closed hyperbolic 3-manifolds do not admit any (topological, hence geometric) canal foliations.
For the entire collection see [Zbl 1481.26002].
Reviewer: Francesco Cattafi (Würzburg)Contact geometry of slant submanifoldshttps://zbmath.org/1491.530022022-09-13T20:28:31.338867ZPublisher's description: This book contains an up-to-date survey and self-contained chapters on contact slant submanifolds and geometry, authored by internationally renowned researchers. The notion of slant submanifolds was introduced by Prof. B. Y. Chen in 1990, and A. Lotta extended this notion in the framework of contact geometry in 1996. Numerous differential geometers have since obtained interesting results on contact slant submanifolds.
The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant \(\xi^\bot\)-, hemi-slant \(\xi^\bot\)-Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric \(f\)-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant submanifolds of almost paracontact manifolds. The book also includes interesting results on slant curves and magnetic curves, where the latter represents trajectories moving on a Riemannian manifold under the action of magnetic field. It presents detailed information on the most recent advances in the area, making it of much value to scientists, educators and graduate students.
The articles of this volume will be reviewed individually.General rotational surfaces satisfying \(\triangle x^T=\varphi x^T\)https://zbmath.org/1491.530082022-09-13T20:28:31.338867Z"Demirbaş, Eray"https://zbmath.org/authors/?q=ai:demirbas.eray"Arslan, Kadri"https://zbmath.org/authors/?q=ai:arslan.kadri"Bulca, Betül"https://zbmath.org/authors/?q=ai:bulca.betulSummary: In the present study we consider rotational surfaces in Euclidean 4-space whose canonical vector field \(x^T\) satisfy the equality \(\triangle x^T=\varphi x^T\). Further, we obtain some results related to three types of general rotational surfaces in \(\mathbb{E}^4\) satisfying this equality. We also give some examples related with these type of surfaces.Constant mean curvature surfaces based on fundamental quadrilateralshttps://zbmath.org/1491.530092022-09-13T20:28:31.338867Z"Bobenko, Alexander I."https://zbmath.org/authors/?q=ai:bobenko.alexander-ivanovich"Heller, Sebastian"https://zbmath.org/authors/?q=ai:heller.sebastian-gregor"Schmitt, Nick"https://zbmath.org/authors/?q=ai:schmitt.nickIn this paper, the authors describe the construction of constant mean curvature (CMC) surfaces with symmetries in Euclidean 3-space and in the round 3-sphere using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.
Reviewer: Atsushi Fujioka (Osaka)Low index capillary minimal surfaces in Riemannian 3-manifoldshttps://zbmath.org/1491.530112022-09-13T20:28:31.338867Z"Longa, Eduardo Rosinato"https://zbmath.org/authors/?q=ai:longa.eduardo-rosinatoIn this work, the author prove a local rigidity result for infinitesimally rigid capillary surfaces in some Riemannian 3-manifolds with mean-convex boundary. After that the author also derive bounds on the genus, number of boundary components and area of any compact two-sided capillary minimal surface with low index under certain assumptions on the curvature of the ambient manifold and of its boundary.
Reviewer: Ameth Ndiaye (Dakar)Semi-symmetric non-metric connections on statistical manifoldshttps://zbmath.org/1491.530172022-09-13T20:28:31.338867Z"Yıldırım, Mustafa"https://zbmath.org/authors/?q=ai:yildirim.mustafaSummary: This study presents a semi-symmetric non-metric connection on statistical manifolds and their submanifolds. The study determines some fundamental properties for the curvature tensor and proves the induced connection on the submanifold with the semi-symmetric non-metric connection. Also, the Gauss, Codazzi, and Ricci equations admitting semi-symmetric non-metric connection have been examined. Finally, a constructive example of such type manifolds is presented.The proportionality principle for Osserman manifoldshttps://zbmath.org/1491.530182022-09-13T20:28:31.338867Z"Andrejić, Vladica"https://zbmath.org/authors/?q=ai:andrejic.vladicaSummary: We give the necessary and sufficient conditions for Jacobi operators that determine an algebraic curvature tensor. This motivates us to introduce the new concept of Jacobi-proportional Riemannian tensors, whose special case is the Rakić duality principle. We prove that all known Osserman tensors (both two-root Osserman and Clifford) are Jacobi-proportional. After the results given by Nikolayevsky, it is known that possible counterexamples of the Osserman conjecture can occur in dimension 16 only, while the reduced Jacobi operator has an eigenvalue of multiplicity 7 or 8. We prove that Jacobi-proportional Osserman tensors that do not satisfy the Osserman conjecture are 2-root with multiplicities 8 and 7, or 3-root with multiplicities 7, 7, and 1.On 2-Stein submanifolds in space formshttps://zbmath.org/1491.530202022-09-13T20:28:31.338867Z"Euh, Yunhee"https://zbmath.org/authors/?q=ai:euh.yunhee"Kim, Jihun"https://zbmath.org/authors/?q=ai:kim.jihun"Nikolayevsky, Yuri"https://zbmath.org/authors/?q=ai:nikolayevsky.yuri"Park, JeongHyeong"https://zbmath.org/authors/?q=ai:park.jeonghyeongSummary: We prove that a 2-stein submanifold in a space form whose normal connection is flat or whose codimension is at most 2, has constant curvature.On submanifolds of the unit sphere with vanishing Möbius form and parallel para-Blaschke tensorhttps://zbmath.org/1491.530212022-09-13T20:28:31.338867Z"Song, Hong Ru"https://zbmath.org/authors/?q=ai:song.hongru"Liu, Xi Min"https://zbmath.org/authors/?q=ai:liu.ximinSummary: The para-Blaschke tensor are extended in this paper from hypersurfaces to general higher codimensional submanifolds in the unit sphere \({\mathbb{S}^n} \), which is invariant under the Möbius transformations on \({\mathbb{S}^n} \). Then some typical new examples of umbilic-free submanifolds in \(\mathbb{S}^n\) with vanishing Möbius form and a parallel para-Blaschke tensor of two distinct eigenvalues, \(D_1\) and \(D_2\), are constructed. The main theorem of this paper is a simple characterization of these newly found examples in terms of the eigenvalues \(D_1\) and \(D_2\).Lie algebras of projective motions of five-dimensional \(H\)-spaces \(H_{221}\) of type \(\{221\}\)https://zbmath.org/1491.530222022-09-13T20:28:31.338867Z"Aminova, A. V."https://zbmath.org/authors/?q=ai:aminova.asya-v|aminova.anya-vasilevna"Khakimov, D. R."https://zbmath.org/authors/?q=ai:khakimov.d-rSummary: We study five-dimensional pseudo-Riemannian \(h\)-spaces \(H_{221}\) of type \(\{221\}\). Necessary and sufficient conditions are determined under which \(H_{221}\) is a space of constant (zero) curvature. Nonhomothetical projective motions in \(H_{221}\) of nonconstant curvature are found, homotheties and isometries of the indicated spaces are investigated. Dimensions, basis elements, and structure equations of maximal projective Lie algebras acting in \(H_{221}\) of nonconstant curvature are determined. As a result, the classification of \(h\)-spaces \(H_{221}\) of type \(\{221\}\) by (non-homothetical) Lie algebras of infinitesimal projective and affine transformations is obtained.Null hypersurfaces in Brinkmann spacetimeshttps://zbmath.org/1491.530232022-09-13T20:28:31.338867Z"Atindogbe, Cyriaque"https://zbmath.org/authors/?q=ai:atindogbe.cyriaque"Mbiakop, Theophile Kemajou"https://zbmath.org/authors/?q=ai:mbiakop.theophile-kemajou(no abstract)Pseudo-symmetric spacetimes admitting \(F(R)\)-gravityhttps://zbmath.org/1491.530242022-09-13T20:28:31.338867Z"De, Uday Chand"https://zbmath.org/authors/?q=ai:de.uday-chand"Altay Demirbag, Sezgin"https://zbmath.org/authors/?q=ai:altay-demirbag.sezgin"Zengin, Füsun Özen"https://zbmath.org/authors/?q=ai:ozen-zengin.fusunSummary: In the first part of the article, the definition of pseudo-symmetric spacetime is given. In the second part, the conditions under which a four-dimensional pseudo-symmetric spacetime has Riemann compatible and Weyl compatible vector fields are obtained, and the spacetime is examined considering the case that this spacetime is a generalized Robertson-Walker spacetime. In the third part of the article, some investigations on four-dimensional pseudo-symmetric spacetimes admitting \(F(R)\)-gravity are made.Almost Hermitian Golden manifoldshttps://zbmath.org/1491.530252022-09-13T20:28:31.338867Z"Bouzir, Habib"https://zbmath.org/authors/?q=ai:bouzir.habib"Beldjilali, Gherici"https://zbmath.org/authors/?q=ai:beldjilali.ghericiSummary: In this paper, we discuss some geometric properties of almost complex Golden structure (i.e. a polynomial structure with the structure polynomial \(Q(X) =X^2-X+\frac{3}{2}I)\) and we introduce such some new classes of almost Hermitian Golden structures. We give a concrete examples.Conformal vector fields on a locally projectively flat Kropina metrichttps://zbmath.org/1491.530272022-09-13T20:28:31.338867Z"Rajeshwari, M. R."https://zbmath.org/authors/?q=ai:rajeshwari.m-r"Narasimhamurthy, S. K."https://zbmath.org/authors/?q=ai:narasimhamurthy.senajji-kamplappaSummary: In this paper, we study and characterize conformal vector fields on a Finsler manifold with the Kropina metric of projectively isotropic flag curvature. Further, we prove that any conformal vector field on a non-Riemannian locally projectively flat Kropina metric of dimension \(n\ge 3\) must be homothetic and completely determine conformal vector fields on a locally projectively flat Kropina metric.Left invariant spray structure on a Lie grouphttps://zbmath.org/1491.530282022-09-13T20:28:31.338867Z"Xu, Ming"https://zbmath.org/authors/?q=ai:xu.mingIn this paper, the author uses the technique of invariant frame to study left-invariant spray structures on a Lie group. He calculates its S-curvature and Riemann curvature, which generalizes L. Huang's formulae in homogeneous Finsler geometry. He proves that using the canonical bi-invariant spray structure as the origin, any left-invariant spray structure can be associated with a spray vector field on the Lie algebra. Further, he finds the correspondence between the geodesics for a left-invariant spray structure and the inverse integral curves of its spray vector field. As an application for this correspondence, he provides an alternative proof of Landsberg Conjecture for homogeneous Finsler surfaces.
Reviewer: Gauree Shanker (Bathinda)Almost Ricci-Bourguignon solitons and geometrical structure in a relativistic perfect fluid spacetimehttps://zbmath.org/1491.530292022-09-13T20:28:31.338867Z"Siddiqui, Aliya Naaz"https://zbmath.org/authors/?q=ai:siddiqui.aliya-naaz"Siddiqi, Mohammed Danish"https://zbmath.org/authors/?q=ai:siddiqi.mohammed-danishThe present study is based on the geometrical bearing of relativistic perfect fluid spacetime and GRW-spacetime in terms of almost Ricci-Bourguignon solitons with torse-forming vector field \(\xi\). A condition for the almost Ricci-Bourguignon solitons to be steady, expanding or shrinking is also given. In particular, when the potential vector field \(\xi\) of the soliton is of gradient type, we derive a Poisson-Laplacian equation from the almost \(\eta \)-Ricci-Bourguignon soliton. Finally, we provide an example of 4-dimensional relativistic spacetime admitting the almost Ricci-Bourguignon and almost \(\eta \)-Ricci-Bourguignon solitons.
Reviewer: Anthony D. Osborne (Keele)Many finite-dimensional lifting bundle gerbes are torsionhttps://zbmath.org/1491.530302022-09-13T20:28:31.338867Z"Roberts, David Michael"https://zbmath.org/authors/?q=ai:roberts.david-michaelSummary: Many bundle gerbes are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson [`A note on bundle gerbes and infinite-dimensionality', \textit{J. Aust. Math. Soc.} \textbf{90}(1)(2011), 81-92] proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion \(DD\)-class. I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles and finite-dimensional twists of topological \(K\)-theory.The holonomy of a singular leafhttps://zbmath.org/1491.530312022-09-13T20:28:31.338867Z"Laurent-Gengoux, Camille"https://zbmath.org/authors/?q=ai:laurent-gengoux.camille"Ryvkin, Leonid"https://zbmath.org/authors/?q=ai:ryvkin.leonid\textit{I. Androulidakis} and \textit{G. Skandalis} [J. Reine Angew. Math. 626, 1--37 (2009; Zbl 1161.53020)] gave a construction for the holonomy groupoid of any singular foliation. Androulidakis and Zambon formulated a holonomy map for singular foliations, which is defined on the holonomy groupoid, rather than the fundamental group of a leaf, as it happens with regular foliations. On the other hand, Laurent-Gengoux, Lavau and Strobl established a universal Lie-\(\infty\) algebroid to every singular foliation.
In the paper under review, the authors construct higher holonomy maps, defined on \(\pi_n(L)\), where \(L\) is a singular leaf \(L\). They take values in the \((n-1)\)-th homotopy group of the universal Lie-\(\infty\) algebroid associated with the transversal foliation to \(L\). Moreover, they show that these holonomy maps form a long exact sequence.
Reviewer: Iakovos Androulidakis (Athína)Some curvature identities on nearly Kähler manifoldshttps://zbmath.org/1491.530322022-09-13T20:28:31.338867Z"Alam, Samser"https://zbmath.org/authors/?q=ai:alam.samser"Bhattacharyya, Arindam"https://zbmath.org/authors/?q=ai:bhattacharyya.arindamSummary: In this paper we have studied and obtained expressions of some curvature identities on nearly Kähler manifold which is con-circularly flat and projectively flat. Also we got interesting results on 6-dimensional nearly Kähler manifold with an example.Some type of semisymmetry on two classes of almost Kenmotsu manifoldshttps://zbmath.org/1491.530332022-09-13T20:28:31.338867Z"Dey, Dibakar"https://zbmath.org/authors/?q=ai:dey.dibakar"Majhi, Pradip"https://zbmath.org/authors/?q=ai:majhi.pradipSummary: The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a \((k, \mu)\)-almost Kenmotsu manifold satisfying the curvature condition \(Q \cdot R = 0\) is locally isometric to the hyperbolic space \(\mathbb{H}^{2n+1}(-1)\). Also in \((k, \mu)\)-almost Kenmotsu manifolds the following conditions: (1) local symmetry \((\nabla R = 0)\), (2) semisymmetry \((R \cdot R = 0)\), (3) \(Q(S, R) = 0\), (4) \(R \cdot R = Q(S, R)\), (5) locally isometric to the hyperbolic space \(\mathbb{H}^{2n+1}(-1)\) are equivalent. Further, it is proved that a \((k, \mu)^\prime\)-almost Kenmotsu manifold satisfying \(Q \cdot R = 0\) is locally isometric to \(\mathbb{H}^{n+1}(-4) \times \mathbb{R}^{n}\) and a \((k, \mu)^\prime\)-almost Kenmotsu manifold satisfying any one of the curvature conditions \(Q(S, R) = 0\) or \(R \cdot R = Q(S, R)\) is either an Einstein manifold or locally isometric to \(\mathbb{H}^{n+1} (-4) \times \mathbb{R}^n\). Finally, an illustrative example is presented.\(J\)-trajectories in 4-dimensional solvable Lie group \(\mathrm{Sol}_0^4\)https://zbmath.org/1491.530342022-09-13T20:28:31.338867Z"Erjavec, Zlatko"https://zbmath.org/authors/?q=ai:erjavec.zlatko"Inoguchi, Jun-ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichiSummary: \(J\)-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation \(\nabla_{\dot{\gamma}}\dot{\gamma}=q J \dot{\gamma}\). In this paper \(J\)-trajectories in the solvable Lie group \(\mathrm{Sol}_0^4\) are investigated. The first and the second curvature of a non-geodesic \(J\)-trajectory in an arbitrary LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic \(J\)-trajectories in \(\mathrm{Sol}_0^4\) are characterized.On the metallic structure on differentiable manifoldshttps://zbmath.org/1491.530352022-09-13T20:28:31.338867Z"Khan, Mohammad Nazrul Islam"https://zbmath.org/authors/?q=ai:khan.mohd-nazrul-islam"Kankarej, Manisha"https://zbmath.org/authors/?q=ai:kankarej.manisha-mSummary: This paper aims to study the metallic structure on a differentiable manifold \(M\) and obtain the metallic structure that acts on complementary distribution \(D_{\mathcal{L}}\) as an almost product structure and complementary distribution \(D_{\mathcal{M}}\) as the null operator. Some calculations on the Nijenhuis tensor of the metallic structure on \(M\) are determined.Invariant sub-manifolds of \(LP\)-Sasakian manifolds with semi-symmetric metric connectionshttps://zbmath.org/1491.530362022-09-13T20:28:31.338867Z"Somashekhara, G."https://zbmath.org/authors/?q=ai:somashekhara.g"Pavani, N."https://zbmath.org/authors/?q=ai:pavani.n"Reddy, P. Siva Kota"https://zbmath.org/authors/?q=ai:reddy.p-siva-kota(no abstract)A remark on CR-structureshttps://zbmath.org/1491.530372022-09-13T20:28:31.338867Z"Verma, Geeta"https://zbmath.org/authors/?q=ai:verma.geeta"Prasad, Kripa Sindhu"https://zbmath.org/authors/?q=ai:prasad.kripasindhuSummary: The present paper aims to establish the relationship between CR-structure and the quadratic structure and find some basic results. Integrability conditions and certain theorems on CR-structure and the quadratic structure are discussed.Integrable LCK manifoldshttps://zbmath.org/1491.530382022-09-13T20:28:31.338867Z"Cappelletti-Montano, Beniamino"https://zbmath.org/authors/?q=ai:cappelletti-montano.beniamino"De Nicola, Antonio"https://zbmath.org/authors/?q=ai:de-nicola.antonio"Yudin, Ivan"https://zbmath.org/authors/?q=ai:yudin.ivanA locally conformal Kähler (LCK) manifold is a Hermitian manifold \((M, J , g)\) whose fundamental 2-form \(\Omega\) and Lee 1-form \(\theta\) satisfy the following identities: \[d\Omega = \theta\wedge \Omega\quad\text{and}\quad d\theta = 0.\] In the present paper, the authors study specifically those LCK manifolds (called integrable) admitting a nowhere zero anti-Lee 1-form \(\eta:=-\theta\circ J\), such that \(\ker\eta\) is an integrable distribution.
As main results, they provide a necessary and sufficient condition for a LCK manifold to be integrable. They study the possibilities that Lee or anti-Lee vector field (defined as the metric duals of \(\theta\) and \(\eta\) respectively) are Killing. Then, in the last sections, they investigate integrable LCK Lie algebras with a particular focus on the unimodular case. The problem of the classification of unimodular integrable LCK Lie algebras is reduced to the classification of pairs of even dimensional matrices satisfying suitable relations involving the complex structure. Examples of both unimodular and non-unimodular integrable LCK Lie algebras are given.
Reviewer: Filippo Salis (Torino)A perturbation approach for Paneitz energy on standard three spherehttps://zbmath.org/1491.530392022-09-13T20:28:31.338867Z"Hang, Fengbo"https://zbmath.org/authors/?q=ai:hang.fengbo"Yang, Paul C."https://zbmath.org/authors/?q=ai:yang.paul-c-pSummary: We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher-order variational problems.On Huber-type theorems in general dimensionshttps://zbmath.org/1491.530402022-09-13T20:28:31.338867Z"Ma, Shiguang"https://zbmath.org/authors/?q=ai:ma.shiguang"Qing, Jie"https://zbmath.org/authors/?q=ai:qing.jieAuthors' abstract: In this paper we present some extensions of the celebrated finite point conformal compactification theorem of \textit{A. Huber} [Comment. Math. Helv. 32, 13--72 (1957; Zbl 0080.15001)] for complete open surfaces to general dimensions based on the \(n\)-Laplace equation in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the improvements of the Arsove-Huber-type theorem on \(n\)-superharmonic functions in our earlier work [Calc. Var. Partial Differ. Equ. 60, No. 6, Paper No. 234, 42 p. (2021; Zbl 1480.31005)]. Moreover, using \(p\)-parabolicity, we push the injectivity theorem of Schoen-Yau to allow some negative curvature and therefore establish the finite point conformal compactification theorem for manifolds which have a conformal immersion into the round sphere. As a side product we establish the injectivity of conformal immersions from \(n\)-parabolicity alone, which is interesting by itself in conformal geometry.
Reviewer: Mohammed El Aïdi (Bogotá)Gradient Einstein-type structures immersed into a Riemannian warped producthttps://zbmath.org/1491.530412022-09-13T20:28:31.338867Z"Batista, Elismar"https://zbmath.org/authors/?q=ai:batista.elismar"Adriano, Levi"https://zbmath.org/authors/?q=ai:adriano.levi-rosa"Tokura, Willian"https://zbmath.org/authors/?q=ai:tokura.willianSummary: In this paper, we study gradient Einstein-type structure immersed into a Riemannian warped product manifold. We obtain some triviality results for the potential function and smooth map \(u\). We investigate conditions for a gradient Einstein-type structure to be totally umbilical, or totally geodesic immersed into a warped product \(I \times_f M^n\). Furthermore, we study rotational hypersurface into \(\mathbb{R} \times_f \mathbb{R}^n\) has a gradient Einstein-type structure.Alexandrov spaces and the Erdős-Perelman problemhttps://zbmath.org/1491.530422022-09-13T20:28:31.338867Z"Che Moguel, Mauricio"https://zbmath.org/authors/?q=ai:che-moguel.mauricioThe seventeenth problem in Erdös' list says the following: If \(S\) is a set of \(2^n+1\) points in \(\mathbb{R}^n\), then 3 of them form an obtuse angle. This problem was solved by Danzer and Grünbaum with an argument of convex geometry. If one thinks of Alexandrov's spaces theory with inferior lower bound curvature, we can focus on extreme sets, and with small ajustements on Danzer and Grünbaum's argument, G. Perelman proved that the extreme points in an Alexandrov Space with non-negative curvature is, at most \(2^n\). This is actually the Erdös-Perelman Problem.
In Section 1 (Erdös Problem: a Proof from ``The Book``) the author gives a short account of the origin of the problem and gives a solution. In Section 2 he reviews some basic elements of the theory of Alexandrov's spaces. In Section 3 the author follows the same approach as in Section 1, now for the Erdös-Perelman problem. In Section 4 (The extreme case of Erdös-Perelman problem), the author is interested in what Alexandrov spaces with the greatest number of extreme points look like. He starts with an example in 2 dimensions and proceeds to the general case. In order to do this the author needs to considers some characteristics of Alexandrov spaces like their stability and quotient by some group actions. This will allow the possibility to count the number of singular orbits of discrete, isometric and cocompact orbits. A compact space \(X\in \mathrm{Alex}^n_0\) that has exactly \(2^n\) extreme points is called a \(n\)-box. Considering precise groups with defined properties the author is able to classify \(n\)-boxes for \(n=1,2,3\).
In the last section (Extreme sets), the author discusses the extreme sets in Alexandrov spaces. Extreme sets generalize the notion of extreme points They also help to show exactly in what way an Alexandrov space is not a Riemannian manifold. The main goal in the author's words is ``\textit{to give some ideas of advanced tools used in Alexandrov spaces theory in order to motivate the reader to study this beautiful branch of metric geometry}''. The author also gives the reader further bibliography for the reader with more interest in further details. He starts with some comments on the family of semi-concave functions and its gradient flows, and explains how these functions are important and how they can be used to define extreme sets. The author gives some examples of extreme sets.
The very last theorem that states that given \(n,k,D\) there exists a constant \(C=C(n,k,D)\) such that for all \(X\in Alex_k^n\) with diam \(X \leq D\), then:
1) the number of extreme sets in \(X\) is at most \(C\),
2) the total Betti number of any extreme set in \(X\) is at most \(C\),
3) the \(m\)-dimensional Hausdorff measure of any extreme \(m\)-dimensional set is at most \(C\).
This last theorem is in fact a generalization of the statement of the Erdös-Perelman Problem
Reviewer: Ana Pereira do Vale (Braga)Bavard's systolically extremal Klein bottles and three dimensional applicationshttps://zbmath.org/1491.530432022-09-13T20:28:31.338867Z"El Mir, Chady"https://zbmath.org/authors/?q=ai:el-mir.chadyThere are 10 equivalence classes of three-dimensional Bieberbach manifolds up to diffeomorphism. (A Bieberbach manifold is a compact manifold that admits a flat Riemannian metric.) Four of these equivalence classes are non-orientable. In this paper, the author constructs on each of these four types of non-orientable Bieberbach 3-manifolds, a two-parameter family of singular metrics which maximize the isosystolic ratio \( \frac{\mathrm{sys}(g)^3}{\mathrm{Vol}(g)}\) in their conformal class. The proof utilizes \textit{C. Bavard}'s construction [Geom. Dedicata 27, No. 3, 349--355 (1988; Zbl 0667.53033)] of a 1-parameter family of singular metrics on the Klein bottle which maximize the isosystolic ratio \(\frac{\mathrm{sys}(g)^2}{\mathrm{Area}(g)}\) in their conformal class.
Reviewer: James Hebda (St. Louis)Distance difference functions on nonconvex boundaries of Riemannian manifoldshttps://zbmath.org/1491.530442022-09-13T20:28:31.338867Z"Ivanov, S. V."https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovich.1|ivanov.sergei-valerevich|ivanov.sergei-vladimirovich|ivanov.sergei-vladimirovich.2|ivanov.sergei-vSuppose \(M\) is a complete Riemannian manifold with non-empty boundary \(F=\partial M\) and with arc-length distance \(d_M:M\times M\rightarrow \mathbb{R}\). For each \(x\in M\), the author considers the distance difference function \(D_x:F\times F\rightarrow \mathbb{R}\) defined by \(D_x(y,z) = d_M(x,y)- d_m(x,z)\), which is then collected into a subset \(\mathcal{D}(M) = \{D_x: x\in M\}\subseteq C(F\times F)\) of the continuous functions on \(F\times F\)
If \((M,g)\) and \((M',g')\) are Riemannian manifolds both with boundary homeomorphic to \(F\), a choice of homeomorphism allows one to view both \(\mathcal{D}(M)\) and \(\mathcal{D}'(M')\) as subsets of \(C(F\times F)\). The main result of the article is that if \(\mathcal{D}(M) = \mathcal{D}'(M')\), then \(M\) and \(M'\) are isometric by an isometry which fixes \(F\) pointwise.
Prior results include establishing an analogous result in the boundaryless case where \(F\) is replaced by an arbitrary open subset of \(M\) [\textit{M. Lassas} and \textit{T. Saksala}, Asian J. Math. 23, No. 2, 173--200 (2019; Zbl 1419.53038); \textit{S. Ivanov}, Geom. Dedicata 207, 167--192 (2020; Zbl 1478.53068)]. In the case of non-empty boundary, an analogous result was established [\textit{M. V. de Hoop} and \textit{T. Saksala}, J. Geom. Anal. 29, No. 4, 3308--3327 (2019; Zbl 1428.53047)] under certain geometric assumptions on \(F\). In the article under review, no assumptions on the boundary are needed.
The idea of the proof is as follows. In [\textit{S. Ivanov}, Geom. Dedicata 207, 167--192 (2020; Zbl 1478.53068)], the author had already established that the map \(\mathcal{D}\) is a locally bi-Lipschitz homeomorphism onto its image. Then, since \(\mathcal{D}(M) = \mathcal{D}'(M')\) one obtains a map \(\phi = (\mathcal{D}')^{-1}\circ \mathcal{D}:M\rightarrow M'\) which is a locally bi-Lipschitz homeomorphism.
The goal then becomes to show that \(\phi\) is a Riemannian isometry. At any point \(p\) where \(\phi\) is differentiable the author establishes this by exploiting the geometry of shortest paths from \(p\) to \(F\). As the set of differentiable, points is dense in \(M\), this is sufficient to prove that \(\phi\) is a metric isometry. Finally, one applies the classical Myers-Steenrod result that a metric isometry is a Riemannian isometry.
Reviewer: Jason DeVito (Martin)Local symmetry rank bound for positive intermediate Ricci curvatureshttps://zbmath.org/1491.530452022-09-13T20:28:31.338867Z"Mouillé, Lawrence"https://zbmath.org/authors/?q=ai:mouille.lawrenceSummary: We use a local argument to prove if an \(r\)-dimensional torus acts isometrically and effectively on a connected \(n\)-dimensional manifold which has positive \(k^{\text{th}}\)-intermediate Ricci curvature at some point, then \(r \leq \lfloor \frac{n+k}{2} \rfloor\). This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.An isoperimetric-type inequality for spacelike submanifold in the Minkowski spacehttps://zbmath.org/1491.530462022-09-13T20:28:31.338867Z"Tsai, Chung-Jun"https://zbmath.org/authors/?q=ai:tsai.chung-jun"Wang, Kai-Hsiang"https://zbmath.org/authors/?q=ai:wang.kai-hsiangSummary: We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.Upper bounds for the first non-zero Steklov eigenvalue via anisotropic mean curvatureshttps://zbmath.org/1491.530472022-09-13T20:28:31.338867Z"Chen, Qun"https://zbmath.org/authors/?q=ai:chen.qun.1|chen.qun"Shi, Jianghai"https://zbmath.org/authors/?q=ai:shi.jianghaiSummary: In this article, we study the eigenvalue problem of the following operator
\[
\begin{cases}
\mathrm{div}\, A\nabla f=0 \quad \text{in } M\\
\frac{\partial f}{\partial v}=pf \quad \text{on }\partial M
\end{cases}
\]
where \(M\) is a compact Riemannian manifold with compact, connected, oriented boundary \(\partial M\), \(A\) denotes a smooth symmetric, positive definite \((1,1)\)-tensor on \(M\). We prove an Reilly-type upper bound for the first non-zero eigenvalue \(p_1\) of this problem via the higher order anisotropic mean curvatures of the boundary \(\partial M\) in \(\mathbb{R}^{n+1}\). Then, we also prove some pinching theorems for \(p_1\). In particular, we give some applications of these results to steklov eigenvalue problem.De Lellis-Topping inequalities on weighted manifolds with boundaryhttps://zbmath.org/1491.530482022-09-13T20:28:31.338867Z"Cruz, F. Jr."https://zbmath.org/authors/?q=ai:cruz.f-jun"Freitas, A."https://zbmath.org/authors/?q=ai:freitas.ana-t|freitas.allan-g|freitas.adelaide-valente|freitas.antonio-a|freitas.augusto-s|freitas.ana-cristina-moreira|freitas.a-g-c|freitas.amauri-a|freitas.a-r-r|freitas.alex-alves|freitas.ayres|freitas.andre|freitas.a-d|freitas.a-b|freitas.ana-p-c"Santos, M."https://zbmath.org/authors/?q=ai:santos.michael-r|santos.marcelino-b|santos.marcelo-f|santos.marta|santos.micael|santos.m-l-o|santos.melina-erica|santos.mauro-lima|santos.marco-antonio-cetale|santos.marcio-c|santos.m-a-f|santos.maria-celia|santos.maria-emma|santos.m-madalena|santos.manoel-j-dos|santos.makson-s|santos.m-franca|santos.manuel-s|santos.m-b-l|santos.marcus-vinicius|santos.marcos-a-c|santos.moises-s|santos.m-t|santos.marta-d|santos.maria-augusta|santos.mauricio-cardoso|santos.mila|santos.marcio-s|santos.maria-luiza-f|santos.maristela-oliveira|santos.mariel|santos.marcelo-m|santos.marcelo-p|santos.maria-r-b|santos.marcelo-r|santos.matheus-c|santos.miguel-aThe De Lellis-Topping inequality implies that if a closed Riemannian manifold with nonnegative Ricci curvature is close to being an Einstein manifold, then its scalar curvature is close to being constant [\textit{C. De Lellis} and \textit{P. M. Topping}, Calc. Var. Partial Differ. Equ. 43, No. 3--4, 347--354 (2012; Zbl 1236.53036)]. In this paper, the authors prove a type of De Lellis-Topping inequality for a symmetric \((0,2)\)-tensor field \(T\) on a compact weighted Riemannian manifold with a convex boundary whose Bakry-Émery Ricci tensor is bounded from below by a negative constant. It is also assumed that the divergence of \(T\) is a constant multiple of \(\nabla(\operatorname{tr}T)\) and that \(T(\nu, -)\) is nonnegative on the boundary, where \(\nu\) is the outward unit normal.
Reviewer: James Hebda (St. Louis)Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifoldshttps://zbmath.org/1491.530492022-09-13T20:28:31.338867Z"Egidi, Michela"https://zbmath.org/authors/?q=ai:egidi.michela"Liu, Shiping"https://zbmath.org/authors/?q=ai:liu.shiping"Münch, Florentin"https://zbmath.org/authors/?q=ai:munch.florentin"Peyerimhoff, Norbert"https://zbmath.org/authors/?q=ai:peyerimhoff.norbertSummary: In this paper, we present a Lichnerowicz-type estimate and (higher order) Buser-type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger-type constants.Ricci flow smoothing for locally collapsing manifoldshttps://zbmath.org/1491.530502022-09-13T20:28:31.338867Z"Huang, Shaosai"https://zbmath.org/authors/?q=ai:huang.shaosai"Wang, Bing"https://zbmath.org/authors/?q=ai:wang.bing.1The theory of Ricci flows is a common tool to regularise Riemannian metrics by replacing the initially given Riemannian metric $g$ with the evolved metric $g(t)$.
To use this tool effectively, it is crucial that the Ricci flow exists for a fixed period of time $T$.
Many results on the existence of a lower bound for $T$ depend on the uniform volume ratio lower bound for the initial metric $g$.
The authors of the current paper provide two theorems for the existence of the Ricci flow up to a fixed period of time without assuming a uniform volume ratio lower bound. The theorems are local in nature and require the Ricci curvature to be bounded from below.
The first theorem is locally modelled on euclidean spaces; the second is locally modeled on flat orbifolds.
As an application, the authors use their Ricci flow smoothing results to detect which collapsing manifolds with Ricci curvature bounded from below are infranil fibre bundles over controlled Riemannian orbifolds.
Reviewer: Thorsten Hertl (Göttingen)A note on the extension of Ricci flowhttps://zbmath.org/1491.530512022-09-13T20:28:31.338867Z"Wu, Guoqiang"https://zbmath.org/authors/?q=ai:wu.guoqiang"Zhang, Jiaogen"https://zbmath.org/authors/?q=ai:zhang.jiaogenSummary: In this paper we study Ricci flow on \(n\) dimensional closed manifold such that the scalar curvature is bounded on \(M\times [0, T)\). We prove that the \(L^\frac{n}{2}\) norm of Riemannian curvature tenor can be controlled by the \(L^\frac{n}{2}\) norm of Weyl curvature tenor. As a corollary, we obtain the Ricci flow can be extended over \(T\) when \(n\) is odd if both the scalar curvature and the \(L^\frac{n}{2}\) norm of Weyl curvature tenor are uniformly bounded.The anisotropic \(p\)-capacity and the anisotropic Minkowski inequalityhttps://zbmath.org/1491.530522022-09-13T20:28:31.338867Z"Xia, Chao"https://zbmath.org/authors/?q=ai:xia.chao"Yin, Jiabin"https://zbmath.org/authors/?q=ai:yin.jiabinSummary: In this paper, we prove a sharp anisotropic \(L^p\) Minkowski inequality involving the total \(L^p\) anisotropic mean curvature and the anisotropic \(p\)-capacity for any bounded domains with smooth boundary in \(\mathbb{R}^n\). As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward \(F\)-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).Billiard trajectories in regular polygons and geodesics on regular polyhedrahttps://zbmath.org/1491.530532022-09-13T20:28:31.338867Z"Fuchs, Dmitry"https://zbmath.org/authors/?q=ai:fuchs.dmitry-bThe paper reports on computer experiments conducted for billiard trajectories in regular polygons and for geodesics on regular polyhedra. The insight won from the experiments are translated in to several conjectures (Conjecture 1.7, 2.3, 2.4, 2.5, 2.6, 2.7, 3.2 in the text). The author expresses his believe that known techniques (e.g. [\textit{C. C. Ward}, Ergodic Theory Dyn. Syst. 18, No. 4, 1019--1042 (1998; Zbl 0915.58059)]) should provide avenues for a verification.
Reviewer: Stefan Suhr (Bochum)Maximal volume entropy rigidity for \(\mathsf{RCD}^\ast (- (N-1), N)\) spaceshttps://zbmath.org/1491.530542022-09-13T20:28:31.338867Z"Connell, Chris"https://zbmath.org/authors/?q=ai:connell.chris"Dai, Xianzhe"https://zbmath.org/authors/?q=ai:dai.xianzhe"Núñez-Zimbrón, Jesús"https://zbmath.org/authors/?q=ai:nunez-zimbron.jesus"Perales, Raquel"https://zbmath.org/authors/?q=ai:perales.raquel"Suárez-Serrato, Pablo"https://zbmath.org/authors/?q=ai:suarez-serrato.pablo"Wei, Guofang"https://zbmath.org/authors/?q=ai:wei.guofangFollowing an ongoing project that extends the results of Ricci curvature bounded below to \(\mathrm{CD}(K,N)\) (\(\mathrm{RCD}(K,N)\) spaces, the authors show that Ledrappier-Wang's maximal volume entropy rigidity [\textit{F. Ledrappier} and \textit{X. Wang}, J. Differ. Geom. 85, No. 3, 461--477 (2010; Zbl 1222.53040)] and Chen-Rong-Xu's almost rigidity result [\textit{L. Chen} et al., J. Differ. Geom. 113, No. 2, 227--272 (2019; Zbl 1430.53054)] also hold for \(\mathrm{RCD}^*(-(N-1),N)\) spaces under an additional condition .
More precisely, the authors define the volume growth entropy of \((X,d, m)\) as
\[
h(X,d,m):=\limsup_{R\to \infty}\frac{1}{R}\mathrm{In}\, \tilde{m}(B_{\tilde{X}}(x,R))
\]
for a compact \(\mathrm{RCD}^*(-(N-1),N)\) space \((X,d, m)\) and its (geometric) universal cover \((\tilde{X}, \tilde{d}, \tilde{m})\). Then they show that \(h(X)\leq N-1\) for \(1<N<\infty\) and the equality holds if and only if \(N\) is an integer and \((\tilde{X}, \tilde{d}, \tilde{m})\) is isomorphic to the \(N\)-dimensional real hyperbolic space \((\mathbb{H}^N, d_{\mathbb{H}^N}, c_1\mathcal{H}^N)\). Here \(c_1\) is a positive constant and \(\mathcal{H}^N)\) is the Hausdorff measure.
Furthermore, let the diameter of \(X\) bounded above by \(D(>0)\) and the first systole of \(X\) bounded blow by \(s(>0)\). The authors show that there exists \(\epsilon(N,s,D)\) such that, if \(h(X)\geq N-1-\epsilon\) for \(0< \epsilon < \epsilon(N,s,D)\), then \(X\) is homeomorphic and \(\Psi(\epsilon|N,s,D)\)-measured Gromov-Hausdorff close to an \(N\)-dimensional hyperbolic manifold.
Noted that the existence of the universal cover of an \(\mathrm{RCD}^*(-(N-1),N)\) space was proved by \textit{A. Mondino} and \textit{G. Wei} [J. Reine Angew. Math. 753, 211--237 (2019; Zbl 1422.53033)]. The compactness of \(X\) is essential since \((0,\infty), |\cdot|, \sinh^{N-1}(x)dx)\) is an \(\mathrm{RCD}^*(-(N-1),N)\) space with volume growth entropy exactly \(N-1\) and it is not the universal cover of a compact \(\mathrm{RCD}^*(-(N-1),N)\) space. On the other hand, the authors \textit{conjecture} that the systole condition is not essential for the last part of the theorem.
Reviewer: Jialong Deng (Beijing)Revisiting linear Weingarten hypersurfaces immersed into a locally symmetric Riemannian manifoldhttps://zbmath.org/1491.530552022-09-13T20:28:31.338867Z"de Lima, Eudes L."https://zbmath.org/authors/?q=ai:de-lima.eudes-leite"de Lima, Henrique F."https://zbmath.org/authors/?q=ai:fernandes-de-lima.henrique"Rocha, Lucas S."https://zbmath.org/authors/?q=ai:rocha.lucas-sSummary: We deal with complete linear Weingarten hypersurfaces immersed into locally symmetric Riemannian manifolds whose sectional curvature obeys certain standard constraints. Under an assumption that such a hypersurface satisfies a suitable Okumura type inequality, we apply a version of the Omori-Yau maximum principle to prove that it must be either totally umbilical or isometric to an isoparametric hypersurface having two distinct principal curvatures. When the ambient space is Einstein, we also use a technique recently developed by \textit{L. J. Alías} and \textit{J. Meléndez} [Mediterr. J. Math. 17, No. 2, Paper No. 61, 14 p. (2020; Zbl 1452.53051)] to establish a sharp integral inequality for compact linear Weingarten hypersurfaces.Rigidity of minimal hypersurfaces with free boundary in a ballhttps://zbmath.org/1491.530562022-09-13T20:28:31.338867Z"Park, Sangwoo"https://zbmath.org/authors/?q=ai:park.sangwoo"Pyo, Juncheol"https://zbmath.org/authors/?q=ai:pyo.juncheolSummary: In this paper, we give two rigidity results of free boundary hypersurfaces in a ball. First, we prove that any minimal hypersurface with free boundary in a closed geodesic ball in a round open hemisphere \(\mathbb{S}^{n+1}_+\) which is Killing-graphical is a geodesic disk. We note that we do not assume any topological condition on the hypersurface. We consider analogous result for self-shrinkers of the mean curvature flow. More precisely, we proved that any graphical self-shrinker with free boundary in a ball centered at the origin in \(\mathbb{R}^{n+1}\) is a flat disk passing through the origin.A note on the classification of noncompact quasi-Einstein manifolds with vanishing condition on the Weyl tensorhttps://zbmath.org/1491.530572022-09-13T20:28:31.338867Z"Baltazar, H."https://zbmath.org/authors/?q=ai:baltazar.halyson-i"Matos Neto, M."https://zbmath.org/authors/?q=ai:matos-neto.mSummary: The aim of this paper is to study complete (noncompact) \(m\)-quasi-Einstein manifolds with \(\lambda =0\) satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an \(m\)-quasi-Einstein manifold (\(m>1\)) with \(\lambda =0\) on a simply connected \(n\)-dimensional manifold \((M^n, g)\), (\(n \geq 4\)), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with \((n-1)\)-dimensional Einstein fiber, provided that \(M\) has fourth-order divergence-free Weyl tensor (i.e. \(div^4W =0\)).Transverse Kähler holonomy in Sasaki geometry and \(\mathcal{S}\)-stabilityhttps://zbmath.org/1491.530582022-09-13T20:28:31.338867Z"Boyer, Charles P."https://zbmath.org/authors/?q=ai:boyer.charles-p"Huang, Hongnian"https://zbmath.org/authors/?q=ai:huang.hongnian"Tønnesen-Friedman, Christina W."https://zbmath.org/authors/?q=ai:tonnesen-friedman.christina-wiisIn this paper the authors study the stability of transverse Kähler holonomy groups in a Sasakian manifold under transverse holomorphic deformations of the Reeb foliation. A Sasakian structure \(\mathcal{S}\) is said to be \(\mathcal{S}\)-stable (or \(\mathcal{S}\)-rigid) if every sufficiently small transverse Kählerian deformation of it remains Sasakian. The main result of this paper states that a Sasakian manifold with vanishing first Betti number and such that the basic Hodge numbers \(h_B^{0,2}=h_B^{2,0}=0\) is \(\mathcal{S}\)-stable. On the other hand, the authors show that a Sasakian manifold with vanishing first Betti number and a compatible irreducible transverse hyperkähler structure is unstable.
Reviewer: Antonio De Nicola (Salerno)Weakly \(\eta\)-Einstein contact manifoldshttps://zbmath.org/1491.530592022-09-13T20:28:31.338867Z"Cho, Jong Taek"https://zbmath.org/authors/?q=ai:cho.jong-taek"Chun, Sun Hyang"https://zbmath.org/authors/?q=ai:chun.sun-hyang"Euh, Yunhee"https://zbmath.org/authors/?q=ai:euh.yunheeSummary: In this paper, we introduce the notion of weakly \(\eta\)-Einstein structure. Then we prove that a 3-dimensional \(\eta\)-Einstein almost contact metric manifold is weakly \(\eta\)-Einstein. Moreover, the generalized Sasakian space forms are weakly \(\eta\)-Einstein. Furthermore, we obtain the characteristic equation for a non-Sasakian contact \((k,\mu)\)-space to be weakly \(\eta\)-Einstein, which provides many interesting examples. In particular, we determine the base manifold whose unit tangent sphere bundle \(T_1 M(c)\) is weakly \(\eta\)-Einstein.Pseudo projective symmetric manifolds and Gray's decompositionhttps://zbmath.org/1491.530602022-09-13T20:28:31.338867Z"De, U. C."https://zbmath.org/authors/?q=ai:de.uday-chand"De, Krishnendu"https://zbmath.org/authors/?q=ai:de.krishnenduSummary: In this paper we characterize pseudo projective symmetric manifolds endowed with the Gray's decomposition. Moreover, we study conformally flat pseudo projective symmetric manifolds and establish that it is a manifold of quasi constant curvature.Geometry of generalized Ricci-type solitons on a class of Riemannian manifoldshttps://zbmath.org/1491.530612022-09-13T20:28:31.338867Z"Kumara, H. Aruna"https://zbmath.org/authors/?q=ai:kumara.huchchappa-aruna"Naik, Devaraja Mallesha"https://zbmath.org/authors/?q=ai:naik.devaraja-mallesha"Venkatesha, V."https://zbmath.org/authors/?q=ai:venkatesha.venkateshaSummary: In this paper, the notion of generalized Ricci-type soliton is introduced and its geometrical relevance on Riemannian \(\mathcal{CR} \)-manifold is established. Particularly, it is shown that a Riemannian \(\mathcal{CR} \)-manifold is Einstein when its metric is a generalized Ricci-type soliton. Next, it has been proved that a Riemannian \(\mathcal{CR} \)-manifold is Einstein-like, when its metric is a generalized gradient Ricci-type almost soliton (or generalized Ricci-type almost soliton for which the soliton vector field is collinear to the \(\mathcal{CR} \)-vector field). Finally, we present an example of generalized Ricci-type solitons which illustrate our results.On irregular Sasaki-Einstein metrics in dimension 5https://zbmath.org/1491.530622022-09-13T20:28:31.338867Z"Süß, H."https://zbmath.org/authors/?q=ai:suss.helmut|suss.hendrikIn the present paper the author studies Fano cone singularities \(X\) polarised by a Reeb field \(\xi\) in the sense of Collins and Sźekelyhidi. The choice of such a polarisation induces a Sasakian metric structure on the link of \(X\). This Sasakian structure is called quasi-regular if the Reeb field generates a one-dimensional torus action; it is called irregular if the corresponding torus is at least two-dimensional. Motivated by questions posed by J. Sparks: ``Are there continuous families of irregular Sasaki-Einstein structures in dimension 5?'' and ``Are there any nontoric irregular Sasaki-Einstein structures in dimension 5?'' and by T. C. Collins, G. Sźekelyhidi such as ``Are there any irregular Sasaki-Einstein structures on \(S^5\)?'', the author proves that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. Also, using K-stability he proves the existence of continuous families of nontoric irregular Sasaki-Einstein structures on odd connected sums of \(S^2 \times S^3\).
Reviewer: Andreas Arvanitoyeorgos (Patras)Heavenly metrics, BPS indices and twistorshttps://zbmath.org/1491.530632022-09-13T20:28:31.338867Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:alexandrov.sergei-yu"Pioline, Boris"https://zbmath.org/authors/?q=ai:pioline.borisSummary: Recently, \textit{T. Bridgeland} [``Geometry from Donaldson-Thomas invariants'', Preprint, \url{arXiv:1912.06504}] defined a complex hyperkähler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function \(W(z,\theta)\) satisfying a heavenly equation, or a potential \(F(z,\theta)\) satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both \(W\) and \(F\) in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of \(F\) as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called \(\tau\) function for arbitrary values of the fiber coordinates \(\theta\), in terms of a suitable two-variable generalization of Barnes' \(G\) function.Spin(7)-manifolds and multisymplectic geometryhttps://zbmath.org/1491.530642022-09-13T20:28:31.338867Z"Kennon, Aaron"https://zbmath.org/authors/?q=ai:kennon.aaronSummary: We utilize Spin(7) identities to prove that the Cayley four-form associated with a torsion-free Spin(7)-structure is non-degenerate in the sense of multisymplectic geometry. Therefore, Spin(7) geometry may be treated as a special case of multisymplectic geometry. We then capitalize on this relationship to make statements about Hamiltonian multivector fields and differential forms associated with torsion-free Spin(7)-structures.
{\copyright 2021 American Institute of Physics}Dirac operators on real spinor bundles of complex typehttps://zbmath.org/1491.530652022-09-13T20:28:31.338867Z"Lazaroiu, C."https://zbmath.org/authors/?q=ai:lazaroiu.calin-iuliu"Shahbazi, C. S."https://zbmath.org/authors/?q=ai:shahbazi.carlos-sLet \(S\) be a real vector bundle over a connected pseudo-Riemannian manifold \((M,g)\) of signature \((p,q)\) and dimension \(d\). A Dirac operator \(D:\Gamma(S) \rightarrow \Gamma(S)\) is a first-order differential operator such that \(D^{2}\) has principal symbol:
\[
\sigma(m,\xi)=g(\xi,\xi)\mathrm{Id}_{S} \ \ \ \ x\in M,\ \ \xi \in T^{*}M \ .
\]
The symbol of \(D\) induces a morphism of bundles \(\gamma:\mathrm{Cl}(M,g) \rightarrow \mathrm{End}(S)\) so that \((S, \gamma)\) becomes a bundle of irreducible real spinors.
Existence of irreducible real spinors on a bundle \(S\) is obstructed. The paper under review shows that, if \(p-q \equiv_{8} 3,7\), a bundle \(S\) has an irreducible Dirac operator if and only if \((M,g)\) admits an adapted \(\mathrm{Spin}^{0}\) structure. This is a principal \(\mathrm{Spin}^{0}_{p,q}\)-bundle \(Q\) over \(M\) endowed with a \(\tilde{\lambda}\)-equivariant map to the orthonormal coframe bundle of \((M,g)\), where
\[
\mathrm{Spin}^{0}_{p,q}=\begin{cases} \mathrm{Spin}_{p,q}\mathrm{Pin}_{0,2} \ \ \ \ \ \ \text{if }p-q \equiv_{8} 3 \\
\mathrm{Spin}_{p,q}\mathrm{Pin}_{2,0} \ \ \ \ \ \ \text{if }p-q \equiv_{8} 7 \end{cases}
\]
and \(\tilde{\lambda}:\mathrm{Spin}^{0}_{p,q}\rightarrow \mathrm{0}(p,q)\) is a certain group homomorphism. \\
The authors characterize pseudo-Riemannian manifolds that admit an adapted \(\mathrm{Spin}^{0}_{p,q}\) structure in terms of the existence of principal \(\mathrm{O}(2)\)-bundles with certain Karoubi Stiefel-Whiteney classes.
Reviewer: Andrea Tamburelli (Houston)Two-component spinorial formalism using quaternions for six-dimensional spacetimeshttps://zbmath.org/1491.530662022-09-13T20:28:31.338867Z"Venâncio, Joás"https://zbmath.org/authors/?q=ai:venancio.joas"Batista, Carlos"https://zbmath.org/authors/?q=ai:batista.carlos-a-sSummary: In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with \(SL(2,\mathbb{H})\), which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of \(SL(2,\mathbb{H})\) do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of \(SO(5, 1)\) is presented, providing a physical interpretation for the elements of \(SL(2,\mathbb{H})\). Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by \(SU(4)\).The stable converse soul question for positively curved homogeneous spaceshttps://zbmath.org/1491.530672022-09-13T20:28:31.338867Z"González-Álvaro, David"https://zbmath.org/authors/?q=ai:gonzalez-alvaro.david"Zibrowius, Marcus"https://zbmath.org/authors/?q=ai:zibrowius.marcusAuthors' abstract: The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E \times \mathbb{R}^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is ``stable under tangential homotopy equivalence'': if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to \(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.
Reviewer: V. V. Gorbatsevich (Moskva)Kac-Moody symmetric spaceshttps://zbmath.org/1491.530682022-09-13T20:28:31.338867Z"Freyn, Walter"https://zbmath.org/authors/?q=ai:freyn.walter"Hartnick, Tobias"https://zbmath.org/authors/?q=ai:hartnick.tobias"Horn, Max"https://zbmath.org/authors/?q=ai:horn.max"Köhl, Ralf"https://zbmath.org/authors/?q=ai:kohl.ralfThe goal of this article of a foundational nature is to establish a general theory of Kac-Moody symmetric spaces over the field \(\mathbb{R}\). The non-Archimedian case is known under the name of ``masures'' or ``hovels'', discovered by \textit{S. Gaussent} and \textit{G. Rousseau} [Ann. Inst. Fourier 58, No. 7, 2605--2657 (2008; Zbl 1161.22007)]. The authors introduce the concept of \emph{topological} symmetric space which is a topological space endowed with symmetries centered at each of his points. These symmetries have to satisfy axioms that where highlighted by \textit{O. Loos} [Bull. Am. Math. Soc. 73, 250--253 (1967; Zbl 0149.41004); Math. Z. 99, 141--170 (1967; Zbl 0148.17403)]: a symmetry has its center as unique (local) fixed point, it is an involution, and there is a kind of associativity axiom. We denote by \(x \cdot y\) the image of \(y\) by the symmetry centered at \(x\).
First, the authors study a general topological symmetric space \(\mathcal X\) and in particular the notion of geodesics on \(\mathcal X\) (which are symmetric subspaces of \(\mathcal X\) which are homeorphic to the symmetric space \(\mathbb{R}\)). They show that one can associated a \(1\)-parameter subgroup of \(\Aut(\mathcal X)\) to any geodesic, and that it enjoys natural properties.
Second, a topological symmetric space is associated to any generalized Cartan matrix \(A\), which is assumed to be irreducible and symmetrizable (or equivalently to any irreducible symmetrizable real Kac-Moody group \(G\)). In the spherical case (when \(A\) corresponds to a finite-dimensional Lie group \(G\)), a symmetric space can be associated considering the set of conjugates of the Cartan involution of \(G\). For general \(A\), the authors consider the same definition, or rather a quotient of it by vector spaces of dimension the corank of \(A\). They define in this way a topological symmetric space denoted \(\overline{\mathcal X}_G\).
One important result is the study of the flats in the symmetric space \(\overline{\mathcal X}_G\). Flats \(F\) are here defined by three axioms: \(F\) is stable under symmetries, \(F\) contains an element \(z\) such that \(z \cdot x = y\) for any \(x,y \in F\), and the commutativity axiom \(x \cdot (z \cdot (y \cdot z)) = y \cdot (z \cdot (x \cdot z))\) holds for any \(x,y,z \in F\). It is shown that \(G\) acts transitively on the set of pairs \((p,F)\), where \(F\) is a flat in \(\overline{\mathcal X}_G\) and \(p \in F\). It follows that all flats are Euclidean of dimension the rank of \(A\). Moreover, the stabilizer in \(G\) of such a pair \((p,F)\) acts on \(F\) and its action is identified with the action of the Weyl group of \(G\).
Geodesic connectedness is also considered: all symmetric spaces \(\overline{\mathcal X}_G\) are geodesically connected, but if \(A\) is not spherical then two general points in \(\overline{\mathcal X}_G\) cannot be connected by only one geodesic.
In the non-spherical and non-affine case, the Tits cone is a non-trivial cone in the Cartan algebra. A global version of this cone is constructed by the authors and named causal structure. It associates to each point \(x\) in \(\overline{\mathcal X}_G\) a subset of \(\overline{\mathcal X}_G\) that intersects any flat containing \(x\) along a cone with tip \(x\), and that is invariant under the automorphism group. This causal structure allows defining the causal boundary, and it is shown among other things that an element in \(G\) is determined by its action on the causal boundary. The causal pre-order it defines is also studied.
Reviewer: Pierre-Emmanuel Chaput (Nancy)Global uniqueness of large stable CMC spheres in asymptotically flat Riemannian \(3\)-manifoldshttps://zbmath.org/1491.530692022-09-13T20:28:31.338867Z"Chodosh, Otis"https://zbmath.org/authors/?q=ai:chodosh.otis"Eichmair, Michael"https://zbmath.org/authors/?q=ai:eichmair.michaelLet \((M,g)\) be a connected, complete Riemannian \(3\)-manifold, \(C^5\)-asymptotic to Schwarzschild, with mass \(m>0\). It is known that the complement of a compact subset of \(M\) admits a foliation by distinguished stable constant mean curvature spheres. The main result of this paper states that if \((M,g)\) is a manifold as before, whose scalar curvature vanishes and with horizon boundary, then every connected, closed, embedded, stable constant mean curvature surface in \((M,g)\), of large enough area, is a leaf of the canonical foliation.
Reviewer: Antonella Nannicini (Firenze)Notes on flat fronts in hyperbolic spacehttps://zbmath.org/1491.530702022-09-13T20:28:31.338867Z"Dubois, J."https://zbmath.org/authors/?q=ai:dubois.jacques-o|dubois.j-m|dubois.jean-guy|dubois.jacques-emile|dubois.jonathan-l|dubois.jean-luc|dubois.jean-emile|dubois.jerome|dubois.joel|dubois.jacque-octave"Hertrich-Jeromin, U."https://zbmath.org/authors/?q=ai:hertrich-jeromin.udo-j"Szewieczek, G."https://zbmath.org/authors/?q=ai:szewieczek.gudrunIn this paper the authors give a short introduction to discrete flat fronts in hyperbolic space and prove that any discrete flat front in the mixed area sense admits a Weierstrass representation.
Reviewer: Ameth Ndiaye (Dakar)Complete spacelike hypersurfaces with constant scalar curvature: descriptions and gapshttps://zbmath.org/1491.530712022-09-13T20:28:31.338867Z"Gervasio Colares, A."https://zbmath.org/authors/?q=ai:colares.antonio-gervasio"de Lima, Eudes L."https://zbmath.org/authors/?q=ai:de-lima.eudes-leite"de Lima, Henrique F."https://zbmath.org/authors/?q=ai:fernandes-de-lima.henriqueThis paper is a study of complete space-like hypersurfaces of constant scalar curvature which are immersed in Lorentzian space forms. The authors obtain sharp inequalities on the total umbilicity tensor of such submanifolds and completely describe the submanifolds that realize these bounds.
Reviewer: James Hebda (St. Louis)A maximum principle for free boundary minimal varieties of arbitrary codimensionhttps://zbmath.org/1491.530722022-09-13T20:28:31.338867Z"Li, Martin Man-chun"https://zbmath.org/authors/?q=ai:li.martin-man-chun"Zhou, Xin"https://zbmath.org/authors/?q=ai:zhou.xin.1Summary: We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.Connected surfaces with boundary minimizing the Willmore energyhttps://zbmath.org/1491.530732022-09-13T20:28:31.338867Z"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Pozzetta, Marco"https://zbmath.org/authors/?q=ai:pozzetta.marcoSummary: For a given family of smooth closed curves \(\gamma^1, \ldots, \gamma^\alpha\subset\mathbb{R}^3\) we consider the problem of finding an elastic \textit{connected} compact surface \(M\) with boundary \(\gamma = \gamma^1\cup \ldots \cup\gamma^\alpha \). This is realized by minimizing the Willmore energy \(\mathcal{W}\) on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is \( 4\pi \), there exists a connected compact minimizer of \(\mathcal{W}\) in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds [\textit{E. Kuwert} and \textit{R. Schätzle}, Ann. Math. (2) 160, No. 1, 315--357 (2004; Zbl 1078.53007); \textit{L. Simon}, Commun. Anal. Geom. 1, No. 2, 281--326 (1993; Zbl 0848.58012)] that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy.Morse index bound for minimal two sphereshttps://zbmath.org/1491.530742022-09-13T20:28:31.338867Z"Sun, Yuchin"https://zbmath.org/authors/?q=ai:sun.yuchinThe author proves that for a closed manifold of dimension at least three, with non-trivial homotopy group \(\pi_3(M)\) and a generic metric, there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width. The collection of finitely many harmonic spheres in the main theorem is constructed by the min-max theory for the energy functional [\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, Geom. Topol. 12, No. 5, 2537--2586 (2008; Zbl 1161.53352)].
Reviewer: Hang Chen (Xi'an)A note about minimal hyperconeshttps://zbmath.org/1491.530752022-09-13T20:28:31.338867Z"Zhang, Yong Sheng"https://zbmath.org/authors/?q=ai:zhang.yongsheng.1|zhang.yongsheng|zhang.yongsheng.2Summary: This short note is concerned with a measure version criterion for hypersurfaces to be minimal. Certain natural flows and associated reflections for many minimal hypercones, including minimal isoparametric hypercones and area-minimizing hypercones, are studied.Ricci solitons on four-dimensional Lorentzian Lie groupshttps://zbmath.org/1491.530762022-09-13T20:28:31.338867Z"Ferreiro-Subrido, M."https://zbmath.org/authors/?q=ai:ferreiro-subrido.m"García-Río, E."https://zbmath.org/authors/?q=ai:garcia-rio.eduardo"Vázquez-Lorenzo, R."https://zbmath.org/authors/?q=ai:vazquez-lorenzo.ramonSummary: We determine all non-Einstein Ricci solitons on four-dimensional Lorentzian Lie groups whose soliton vector field is left-invariant. In addition to pp-wave and plane wave Lie groups, there are four families of Lorentzian metrics on semi-direct extensions \(\mathbb{R}^3\rtimes \mathbb{R}\) and \(E(1, 1)\rtimes \mathbb{R}\). We show that some of these Ricci solitons are conformally Einstein and they may be expanding, steady or shrinking.Twisting non-shearing congruences of null geodesics, almost CR structures and Einstein metrics in even dimensionshttps://zbmath.org/1491.530772022-09-13T20:28:31.338867Z"Taghavi-Chabert, Arman"https://zbmath.org/authors/?q=ai:taghavi-chabert.armanThis paper is devoted to the study of the geometry of twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature.
The author gives a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, i.e., such that the screen bundle of the congruence is equipped with a bundle complex structure. In this case the leaf space of the congruence locally acquires a partially integrable contact almost CR structure of positive definite signature. Further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure and for the flatness of the latter are given. It is shown that under certain natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR-Einstein structure on the leaf space of the congruence. These metrics depend on three paramenters and include the Feffermann-Einstein metric and Taub-NUT-(A)dS metric in the integrable case. In the non-integrable case, new solutions to the Einstein field equations are obtained. These can be constructed from strictly almost Kähler-Einstein manifolds.
Reviewer: María Ferreiro-Subrido (Santiago de Compostela)A note on Griffiths' conjecture about the positivity of Chern-Weil formshttps://zbmath.org/1491.530782022-09-13T20:28:31.338867Z"Fagioli, Filippo"https://zbmath.org/authors/?q=ai:fagioli.filippoLet \((E, h)\) be a Griffiths semipositive Hermitian holomorphic vector bundle of rank 3 over a complex manifold. In this paper, the author proves the positivity of the characteristic differential form
\[
c_1(E, h)\wedge c_2(E, h) -c_3(E, h).
\]
Here, for \(k=1,2,3\), \(c_k(E,h)\) denotes the Chern form of bidegree \((k,k)\), which represents the Chern class \(c_k(E)\) of the vector bundle \(E\). This provides a new evidence towards Griffiths' conjecture about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, the author establishes the following new chain of inequalities between Chern forms
\[
c_1(E, h)^3 \ge c_1(E, h) \wedge c_2(E, h) \ge c_3(E, h).
\]
The author also shows how to obtain the positivity of the second Chern form \(c_2(E, h)\) in any rank, if \((E, h)\) is Griffiths positive. This is obtained by adapting the Griffiths' result on the positivity of \(c_2(E, h)\) in rank 2.
The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.
Reviewer: Riccardo Piovani (Parma)On the squares of umbilicity factors of null hypersurfaces in indefinite Kähler manifoldshttps://zbmath.org/1491.530792022-09-13T20:28:31.338867Z"Ssekajja, Samuel"https://zbmath.org/authors/?q=ai:ssekajja.samuelLet \((\overline{M}, \overline{g}, \overline{J})\) be an indefinite Kaehler manifold. Any null hypersurface \((M,g)\) of \((\overline{M}, \overline{g}, \overline{J})\) is endowed with two special null vector fields \(U := -\overline{J} N\) and \(V := -\overline{J} \xi\), where \(N\) and \(\xi\), respectively, span the transversal and normal bundle (radical distribution) to \(\overline{M}\). This paper defines the \(U\) and \(V\)-null sectional curvatures, analogously to the null sectional curvature defined by \textit{S. G. Harris} [Indiana Univ. Math. J. 31, 289--308 (1982; Zbl 0496.53042)], and obtains several results for totally umbilic (resp.~screen totally umbilic) null hypersurfaces.
Reviewer: Yoshinori Hashimoto (Osaka)A new characterization of Kenmotsu manifolds with respect to \(Q\) tensorhttps://zbmath.org/1491.530822022-09-13T20:28:31.338867Z"Yıldırım, Mustafa"https://zbmath.org/authors/?q=ai:yildirim.mustafaSummary: This paper introduces a new characterization of Kenmotsu manifolds and analyzes curvature conditions of Kenmotsu manifolds admitting the \(Q\) tensor whose trace is the well-known \(Z\) tensor. New types of Kenmotsu manifold are defined which are named \(\phi - Q\) symmetric Kenmotsu manifold and \(\phi - Q\) recurrent Kenmotsu manifold. Various properties of such a \(2 m + 1\)-dimensional Kenmotsu manifold are studied. A theorem is given which states that a three-dimensional locally \(\phi - Q\) recurrent and locally \(\phi - Q\) symmetric Kenmotsu manifold admitting \(Q\) tensor is a manifold of constant curvature. Some examples of locally \(\phi - Q\) symmetric Kenmotsu manifolds and Kenmotsu manifolds admitting \(Q\) tensor are provided.Sasakian 3-metric as a generalized Ricci-Yamabe solitonhttps://zbmath.org/1491.530842022-09-13T20:28:31.338867Z"Dey, Dibakar"https://zbmath.org/authors/?q=ai:dey.dibakar"Majhi, Pradip"https://zbmath.org/authors/?q=ai:majhi.pradipSummary: In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if \((g, V, \lambda, \alpha, \beta, \gamma)\) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold \(M\) with potential function \(f\), then \(M\) is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field \(V\) is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field \(\xi\). Further, if \(h\) is the Hodge-de Rham potential for \(V\), then, upto a constant, \(f = h\).Generalized connections, spinors, and integrability of generalized structures on Courant algebroidshttps://zbmath.org/1491.530862022-09-13T20:28:31.338867Z"Cortés, Vicente"https://zbmath.org/authors/?q=ai:cortes.vicente"David, Liana"https://zbmath.org/authors/?q=ai:david.lianaSummary: We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian structures \((G, \mathcal J)\) and generalized almost hyper-Hermitian structures \((G, \mathcal J_1, \mathcal J_2, \mathcal J_3)\) defined on a regular Courant algebroid \(E\) in terms of canonically defined differential operators on spinor bundles associated to \(E_{\pm}\) (the subbundles of \(E\) determined by the generalized metric \(G\)).Self-similar curve shortening flow in hyperbolic 2-spacehttps://zbmath.org/1491.531002022-09-13T20:28:31.338867Z"Woolgar, Eric"https://zbmath.org/authors/?q=ai:woolgar.eric"Xie, Ran"https://zbmath.org/authors/?q=ai:xie.ranThe authors study the geometric flow
\[ \partial X/\partial t = \kappa_g\cdot N,\tag{\(\ast\)}\]
where \(\kappa_g\) denotes the geodesic curvature, acting in the space of curves on a surface \(M\subset \mathbb{R}^3\). The main result is formulated in the following theorem.
\textbf{Theorem}: Let \(\mathbb M^3\) be \(\mathbb R^3\) equipped with the quadratic form (Minkowski metric) \(\eta = \operatorname{diag} (1, 1, -1)\) and denote the Minkowski inner product by \(\langle U,W \rangle_\eta = \eta (U,W)\) for \(U,W\in \mathbb M^3\). For each \(v\in \mathbb{M}^3 \smallsetminus \{ 0\}\) there is a 2-parameter family of nontrivial solutions X of the flow (\(\ast\)) evolving by isometries in the standard hyperbolic plane \(\mathbb H^2\). These soliton curves are complete, unbounded, and properly embedded, and are asymptotic either to a horocycle or to a geodesic. The soliton is not asymptotic to a geodesic
\begin{itemize}
\item[(i)] at both ends, or
\item[(ii)] if \(v\) is time-like, or
\item[(iii)] if \(\mu (s) = \langle X(s), v\rangle_\eta\) has a critical point (and it can have at most one critical point), or
\item[(iv)] if \(\mu (s)\) has a zero (and it can have at most one zero).
\end{itemize}
Several applications of this result are provided.
Reviewer: Paweł Walczak (Łódź)What does a vector field know about volume?https://zbmath.org/1491.570272022-09-13T20:28:31.338867Z"Geiges, Hansjörg"https://zbmath.org/authors/?q=ai:geiges.hansjorgA vector field \(X\) without zeros on a manifold \(M\) is called geodesible if there is a Riemannian metric on \(M\) such that the flow lines of \(X\) are unit speed geodesics -- equivalently, if there exists a one-form \(\alpha\) invariant under the flow of \(X\) such that \(\alpha(X) = 1\). It was observed by \textit{C. B. Croke} and \textit{B. Kleiner} [J. Differ. Geom. 39, No. 3, 659--680 (1994; Zbl 0807.53035)] that if \(M^{2n+1}\) is closed and orientable, the integral \(\int_M \alpha\wedge (d\alpha)^n\) is independent of the form \(\alpha\); in the paper at hand this number is hence called the volume of \(X\). In particular, the volume of a closed contact manifold is determined by the Reeb vector field alone. This raised the question, posed to the author by Claude Viterbo, whether there exist nondiffeomorphic contact forms on the same manifold with the same Reeb vector field. This paper contains a variety of results motivated by this question (which is answered in the affirmative by giving explicit examples as Boothby-Wang bundles, using a construction of \textit{D. McDuff} [Invent. Math. 89, 13--36 (1987; Zbl 0625.53040)] of cohomologous, non-diffeomorphic symplectic forms) as well as the above notion of volume. Amongst others, the question whether one can compute the volume of a geodesible vector field explicitly from the vector field alone is answered positively in some special cases, such as Seifert fibered \(3\)-manifolds. Moreover, theorems of Gauß-Bonnet and Poincaré-Hopf are proved for \(2\)-dimensional orbifolds.
Reviewer: Oliver Goertsches (Marburg)Prolongations of golden structure to bundles of infinitely near pointshttps://zbmath.org/1491.580012022-09-13T20:28:31.338867Z"Nono, Georges Florian Wankap"https://zbmath.org/authors/?q=ai:nono.georges-florian-wankap"Ntyam, Achille"https://zbmath.org/authors/?q=ai:ntyam.achille"Mang-Massou, Emmanuel Hinamari"https://zbmath.org/authors/?q=ai:mang-massou.emmanuel-hinamariSummary: For a golden-structure \(\zeta\) on a smooth manifold \(M\) and any covariant functor which assigns to \(M\) its bundle \(M^A\) of infinitely near points of \(A\)-king, we define the golden structure \(\zeta^A\) on \(M^A\) and prove that \(\zeta\) is integrable if and only if so is \(\zeta^A\). We also investigate the integrability, parallelism, half parallelism and anti-half parallelism of the golden-structure \(\zeta^A\) and their associated distributions on \(M^A\).The measurement and analysis of shapes. An application of hydrodynamics and probability theoryhttps://zbmath.org/1491.580022022-09-13T20:28:31.338867Z"Benn, James"https://zbmath.org/authors/?q=ai:benn.james"Marsland, Stephen"https://zbmath.org/authors/?q=ai:marsland.stephen-rSummary: A de Rham \(p\)-current can be viewed as a map (the current map) between the set of embeddings of a closed \(p\)-dimensional manifold into an ambient \(n\)-manifold and the set of linear functionals on differential \(p\)-forms. We demonstrate that, for suitably chosen Sobolev topologies on both the space of embeddings and the space of \(p\)-forms, the current map is continuously differentiable, with an image that consists of bounded linear functionals on \(p\)-forms. Using the Riesz representation theorem, we prove that each \(p\)-current can be represented by a unique co-exact differential form that has a particular interpretation depending on \(p\). Embeddings of a manifold can be thought of as shapes with a prescribed topology. Our analysis of the current map provides us with representations of shapes that can be used for the measurement and statistical analysis of collections of shapes. We consider two special cases of our general analysis and prove that: (1) if \(p=n-1\) then closed, embedded, co-dimension one surfaces are naturally represented by probability distributions on the ambient manifold and (2) if \(p=1\) then closed, embedded, one-dimensional curves are naturally represented by fluid flows on the ambient manifold. In each case, we outline some statistical applications using an \({\dot{H}}^1\) and \(L^2\) metric, respectively.The spectrum of the Laplacian and volume growth of proper minimal submanifoldshttps://zbmath.org/1491.580062022-09-13T20:28:31.338867Z"Bessa, G. Pacelli"https://zbmath.org/authors/?q=ai:pacelli-bessa.gregorio"Gimeno, Vicent"https://zbmath.org/authors/?q=ai:gimeno.vicent"Polymerakis, Panagiotis"https://zbmath.org/authors/?q=ai:polymerakis.panagiotisSummary: We give upper bounds for the bottom of the essential spectrum of properly immersed minimal submanifolds of \(\mathbb{R}^n\) in terms of their volume growth. Our result can be viewed as an extrinsic version of \textit{R. Brooks}'s essential spectrum estimate [Math. Z. 178, 501--508 (1981; Zbl 0458.58024)] and it gives a fairly general answer to a question of \textit{S. T. Yau} [Asian J. Math. 4, No. 1, 235--278 (2000; Zbl 1031.53004)] about upper bounds for the first eigenvalue (bottom of the spectrum) of immersed minimal surfaces of \(\mathbb{R}^3\).Approximations of the connection Laplacian spectrahttps://zbmath.org/1491.580072022-09-13T20:28:31.338867Z"Burago, Dmitri"https://zbmath.org/authors/?q=ai:burago.dmitri"Ivanov, Sergei"https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovich"Kurylev, Yaroslav"https://zbmath.org/authors/?q=ai:kurylev.yaroslav-v"Lu, Jinpeng"https://zbmath.org/authors/?q=ai:lu.jinpengBriefly, the authors focus their study on a convolution-type operator on vector bundles over metric-measure spaces. Also, they state that the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.
Reviewer: Mohammed El Aïdi (Bogotá)Intrinsic dimension of geometric data setshttps://zbmath.org/1491.681882022-09-13T20:28:31.338867Z"Hanika, Tom"https://zbmath.org/authors/?q=ai:hanika.tom"Schneider, Friedrich Martin"https://zbmath.org/authors/?q=ai:schneider.friedrich-martin"Stumme, Gerd"https://zbmath.org/authors/?q=ai:stumme.gerdFollowing \textit{V. Pestov}'s suggestions in [``Intrinsic dimension of a dataset: what properties does one expect?'', in: Proceedings of the international joint conference on neural networks, IJCNN 2007, celebrating 20 years of neural networks, Orlando, Florida, USA, August 12--17, 2007, 2959--2964 (2007)], the authors extend Pestov's results in [\textit{V. Pestov}, Inf. Process. Lett. 73, No. 1--2, 47--51 (2000; Zbl 1339.68245); Neural Netw. 21, No. 2--3, 204--213 (2008; Zbl 1254.68102)] to geometric data sets.
A geometric data set is a triple \(D = (X, F, \mu)\) consisting of a set \(X\) equipped with a tame set \(F\subset \mathbb{R}^X\) such that \((X, d_F)\) is a separable complete metric space, and a probability measure \(\mu\) with full support on the Borel \(\sigma\)-algebra of \((X, d_F)\). Here \(d_F(x, y):=\sup\{|f(x)-f(y) \mid f\in F\}\) and the set of functions \(F\) is called tame if \(d_F(x,y)\leq \infty\) for all \(x, y \in X\). After showing that observable distance is a metric on the set of isomorphic geometric data sets, the authors consider the concentration and observable diameters of data.
In the last two parts of the paper, the authors define a dimension function as an axiomatic approach to intrinsic dimension of geometric data sets and compute the dimension function for data sets in \(\mathbb{R}^n\) and data sets resembling incidence structures.
Reviewer: Jialong Deng (Beijing)Infrared scaling for a graviton condensatehttps://zbmath.org/1491.830042022-09-13T20:28:31.338867Z"Bose, Sougato"https://zbmath.org/authors/?q=ai:bose.sougato"Mazumdar, Anupam"https://zbmath.org/authors/?q=ai:mazumdar.anupam"Toroš, Marko"https://zbmath.org/authors/?q=ai:toros.markoSummary: The coupling between gravity and matter provides an intriguing length scale in the infrared for theories of gravity within Einstein-Hilbert action and beyond. In particular, we will show that such an infrared length scale is determined by the number of gravitons \(N_g \gg 1\) associated to a given mass in the non-relativistic limit. After tracing out the matter degrees of freedom, the graviton vacuum is found to be in a displaced vacuum with an occupation number of gravitons \(N_g \gg 1\). In the infrared, the length scale appears to be \(L = \sqrt{N_g}\ell_p\), where \(L\) is the new infrared length scale, and \(\ell_p\) is the Planck length. In a specific example, we have found that the infrared length scale is greater than the Schwarzschild radius for a slowly moving in-falling thin shell of matter. We will argue that the appearance of such an infrared length scale in higher curvature theories of gravity, such as in quadratic and cubic curvature theories of gravity, is also expected. Furthermore, we will show that gravity is fundamentally different from the electromagnetic interaction where the number of photons, \(N_p\), is the \textit{fine structure constant} after tracing out an electron wave function.Primordial black holes and gravitational waves from parametric amplification of curvature perturbationshttps://zbmath.org/1491.830092022-09-13T20:28:31.338867Z"Cai, Rong-Gen"https://zbmath.org/authors/?q=ai:cai.ronggen"Guo, Zong-Kuan"https://zbmath.org/authors/?q=ai:guo.zong-kuan"Liu, Jing"https://zbmath.org/authors/?q=ai:liu.jing.1|liu.jing"Liu, Lang"https://zbmath.org/authors/?q=ai:liu.lang"Yang, Xing-Yu"https://zbmath.org/authors/?q=ai:yang.xingyu(no abstract)Generation of primordial black holes and gravitational waves from dilaton-gauge field dynamicshttps://zbmath.org/1491.830112022-09-13T20:28:31.338867Z"Kawasaki, Masahiro"https://zbmath.org/authors/?q=ai:kawasaki.masahiro"Nakatsuka, Hiromasa"https://zbmath.org/authors/?q=ai:nakatsuka.hiromasa"Obata, Ippei"https://zbmath.org/authors/?q=ai:obata.ippei(no abstract)Existence of new singularities in Einstein-aether theoryhttps://zbmath.org/1491.830182022-09-13T20:28:31.338867Z"Chan, R."https://zbmath.org/authors/?q=ai:chan.raymond-hon-fu|chan.roy|chan.ringo|chan.roberto|chan.roath|chan.ray"da Silva, M. F. A."https://zbmath.org/authors/?q=ai:da-silva.m-f-a"Satheeshkumar, V. H."https://zbmath.org/authors/?q=ai:satheeshkumar.v-h(no abstract)Stochastic axion dark matter in axion landscapehttps://zbmath.org/1491.830242022-09-13T20:28:31.338867Z"Nakagawa, Shota"https://zbmath.org/authors/?q=ai:nakagawa.shota"Takahashi, Fuminobu"https://zbmath.org/authors/?q=ai:takahashi.fuminobu"Yin, Wen"https://zbmath.org/authors/?q=ai:yin.wen(no abstract)Generalization of the 2-form interactionshttps://zbmath.org/1491.830382022-09-13T20:28:31.338867Z"Heisenberg, Lavinia"https://zbmath.org/authors/?q=ai:heisenberg.lavinia"Trenkler, Georg"https://zbmath.org/authors/?q=ai:trenkler.georg(no abstract)Superconformal generalizations of auxiliary vector modified polynomial \(f(R)\) theorieshttps://zbmath.org/1491.830492022-09-13T20:28:31.338867Z"Boran, Sibel"https://zbmath.org/authors/?q=ai:boran.sibel"Kahya, Emre Onur"https://zbmath.org/authors/?q=ai:kahya.emre-onur"Ozdemir, Nese"https://zbmath.org/authors/?q=ai:ozdemir.nese"Ozkan, Mehmet"https://zbmath.org/authors/?q=ai:ozkan.mehmet"Zorba, Utku"https://zbmath.org/authors/?q=ai:zorba.utku(no abstract)Testing kinetically coupled inflation models with CMB distortionshttps://zbmath.org/1491.830512022-09-13T20:28:31.338867Z"Dai, Rui"https://zbmath.org/authors/?q=ai:dai.rui"Zhu, Yi"https://zbmath.org/authors/?q=ai:zhu.yi.1|zhu.yi|zhu.yi.3|zhu.yi.2(no abstract)On the slope of the curvature power spectrum in non-attractor inflationhttps://zbmath.org/1491.830532022-09-13T20:28:31.338867Z"Özsoy, Ogan"https://zbmath.org/authors/?q=ai:ozsoy.ogan"Tasinato, Gianmassimo"https://zbmath.org/authors/?q=ai:tasinato.gianmassimo(no abstract)Destabilization of the EW vacuum in non-minimally coupled inflationhttps://zbmath.org/1491.830542022-09-13T20:28:31.338867Z"Rusak, Stanislav"https://zbmath.org/authors/?q=ai:rusak.stanislav(no abstract)