Recent zbMATH articles in MSC 53https://zbmath.org/atom/cc/532022-11-17T18:59:28.764376ZUnknown authorWerkzeugFrom Barth sextics to minimal surfaces: printing mathematics in 3 dimensionshttps://zbmath.org/1496.000122022-11-17T18:59:28.764376Z"Hershberger, Scott"https://zbmath.org/authors/?q=ai:hershberger.scott(no abstract)Structural models based on minimal surfaces and geodesicshttps://zbmath.org/1496.000502022-11-17T18:59:28.764376Z"Gonzalez-Quintial, Francisco"https://zbmath.org/authors/?q=ai:gonzalez-quintial.francisco"Martin-Pastor, Andres"https://zbmath.org/authors/?q=ai:martin-pastor.andresSummary: This article presents the results of research carried out with respect to the geometric, formal and structural adaptation of minimal surfaces. These surfaces were discretized into strips developable on geodesic curves, and then used for the construction of timber gridshells. For this project, both physical and virtual models derived from the same geometric models were used. The objective was to demonstrate the validity of the use of models and the transformation that modelling is undergoing due to the use of digital tools, both software and hardware. These include, on the one hand, drawing and analysis software and, on the other, digitally controlled fabrication tools. This research focuses specifically on the design and construction of the Scherk Pavilion, a space where the results of various experiments in which the common factor was the use of models was transferred to a real scale.Geometry of curves and surfaces in contemporary chair designhttps://zbmath.org/1496.000512022-11-17T18:59:28.764376Z"Lastra, Alberto"https://zbmath.org/authors/?q=ai:lastra.alberto"De Miguel, Manuel"https://zbmath.org/authors/?q=ai:de-miguel.manuelSummary: In the present work, we focus on some of the current trends used in furniture design, from a dual point of view: differential geometry of curves and surfaces, and the existing perspective deriving from the usual techniques of computer-aided design. The contributions of architects such as Alvar Aalto, Mies van der Rohe, Marcel Breuer, Arne Jacobsen and Charles and Ray Eames to contemporary chair design are related to these techniques. Among them, we point out those which are performed by means of spatial geometric transformations of curves and surfaces, with an emphasis on ruled surfaces.Discrete Ricci curvature-based statistics for soft setshttps://zbmath.org/1496.032092022-11-17T18:59:28.764376Z"Akgüller, Ömer"https://zbmath.org/authors/?q=ai:akguller.omerSummary: Soft sets are efficient mathematical structures to model systems in multiple relations. Since a soft set is basically set system, it is possible to endow them with a proper distance function to obtain a metric space. By this embedding, we propose a discretization of the Ricci curvatures that stresses the relational character of universe elements in a soft set through the analysis of parameters rather than the elements themselves. The Forman and Ollivier-type Ricci curvatures we propose here quantifies the trade-off between parameter size and the cardinality of participation of parameterized universe elements in other parameters. Such discretizations of the Ricci curvature have already been applied to complex systems; however, it has not yet been formulated for soft sets. In this study, our main question is whether the defined geometric concept determines statistics for soft set models. Two examples are discussed for the answer to this question. The first example Ricci on soft sets model of occupational accidents occurred in Turkey in 2013--2014 is compared with the Wasserstein distance of the curvature distributions. The second example is the use of Ricci curvatures as an indicator in the soft sets model of a financial system while the system is in stress. These real world examples show that discrete Ricci curvatures for soft sets offer effective statistics.Distance invariant method for normalization of indexed differentialshttps://zbmath.org/1496.130412022-11-17T18:59:28.764376Z"Liu, Jiang"https://zbmath.org/authors/?q=ai:liu.jiang"Ni, Feng"https://zbmath.org/authors/?q=ai:ni.fengThe paper under review is devoted to the problems of symbolic manipulation of indexed expressions (e.g., tensor expressions). The paper introduces a concept of distance from free indices to dummy indices and shows that it is an invariant with respect to certain symmetries. Then the authors present an index replacement algorithm which is uniquely determined by the distance, and use it to develop a normalization algorithm with respect to monoterm symmetries for polynomials in the ring of all partial differential polynomials with Einstein multiplication. The complexity is at most \(O(m^{2})\), lower than known algorithms, where \(m\) is the sum of the number of free indices and the number of pairs of dummy indices. Furthermore, using the method of index replacement, and by choosing the monomial associated with the smallest numerical list as the canonical form, a normalization algorithm is provided for partial differential polynomials with Einstein multiplication of order \(\leq 2\), which is independent of function classifications.
Reviewer: Alexander B. Levin (Washington)Recent progress in mathematicshttps://zbmath.org/1496.140032022-11-17T18:59:28.764376ZPublisher's description: This book consists of five chapters presenting problems of current research in mathematics, with its history and development, current state, and possible future direction. Four of the chapters are expository in nature while one is based more directly on research. All deal with important areas of mathematics, however, such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. This book is addressed to researchers who are interested in those subject areas.
Young-Hoon Kiem discusses classical enumerative geometry before string theory and improvements after string theory as well as some recent advances in quantum singularity theory, Donaldson-Thomas theory for Calabi-Yau 4-folds, and Vafa-Witten invariants.
Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classical local well-posedness results, Beale-Kato-Majda's blow-up criterion, and recent studies on the singularity problem for the 2D Boussinesq equations.
Simon Brendle discusses recent developments that have led to a complete classification of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3.
Hyeonbae Kang reviews some of the developments in the Neumann-Poincare operator (NPO). His topics include visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO.
Danny Calegari provides an explicit description of the shift locus as a complex of spaces over a contractible building. He describes the pieces in terms of dynamically extended laminations and of certain explicit ``discriminant-like'' affine algebraic varieties.
The articles of this volume will be reviewed individually.Constrained systems, generalized Hamilton-Jacobi actions, and quantizationhttps://zbmath.org/1496.140152022-11-17T18:59:28.764376Z"Cattaneo, Alberto S."https://zbmath.org/authors/?q=ai:cattaneo.alberto-sergio"Mnev, Pavel"https://zbmath.org/authors/?q=ai:mnev.pavel"Wernli, Konstantin"https://zbmath.org/authors/?q=ai:wernli.konstantinThis paper considers the situation of 1-dimensional field theories from the viewpoint of perturbative gauge theories on manifolds with boundary, hence where the system has certain constraints. In particular, it considers properties of the Hamilton-Jacobi action within the perturbative quantization setting of the BV-BFV formalism, which is a gauge formlism for manifolds with boundary (compatible with cutting-gluing) developed recently by the first two authors together with \textit{N. Reshetikhin} in [Commun. Math. Phys. 332, No. 2, 535--603 (2014; Zbl 1302.81141); Commun. Math. Phys. 357, No. 2, 631--730 (2018; Zbl 1390.81381)]. Since they are considering the case of 1-dimensional source manifolds in this paper, the BFV formalism is applied to the endpoints (which is, in this case, the boundary of the 1-dimensional source). An interesting result of this paper is included in the explicit computation of examples, namely that the toy model for nonabelian Chern-Simons theory and the toy model for 7D Chern-Simons theory endowed with nonlinear Hitchin Polarization do not have quantum corrections in the physical part. Furthermore, they provide a concise collection of background material for the sake of completeness and better understanding.
Reviewer: Nima Moshayedi (Zürich)A criterion for the existence of logarithmic connections on curves over a perfect fieldhttps://zbmath.org/1496.140322022-11-17T18:59:28.764376Z"Manikandan, S."https://zbmath.org/authors/?q=ai:manikandan.sreekanth-k|manikandan.sreenath-k"Singh, Anoop"https://zbmath.org/authors/?q=ai:singh.anoopThe goal of this article is to find criterion for the existence of logarithmic connections over a smooth, irreducible projective curve \(X\) over a perfect field \(k\). The field \(k\) need not be of characteristic zero or algebraically closed. Let \(E\) be a vector bundle on \(X\). Choose a set \(S\) of distinct points on \(X\). Given a point \(x \in X\), an endomorphism of the fiber \(E(x)\) of the vector bundle \(E\) is called \emph{rigid} if it commutes with every global endomorphism of \(E\). For every point \(x \in S\), fix rigid endomorphisms \(A(x) \in \mathrm{End}(E(x))\). It is known that if \(k\) is algebraically closed of characteristic zero, then \(E\) admits a logarithmic connection singular over \(S\) with residue \(A(x)\) for every \(x \in S\) if and only if for every indecomposable component \(F\) of \(E\), the degree of \(F\) equals the negative of the sum over all points \(x\) in \(S\), of the trace of the endomorphism \(A(x)\) restricted to \(F(x)\). The authors prove that if \(k\) is instead of characteristic \(p\), then \(E\) admits such a logarithmic connection (i.e., singular over \(S\) with residues \(A(x)\)) if and only if the same condition holds for the degree of every indecomposable components of \(E\), modulo \(p\). Furthermore, if \(k\) is a perfect field but not necessarily algebraically closed, then the same equivalent condition for the existence of logarithmic connections hold after an additional assumption: the rank of every indecomposable component of \(E\) is not divisible by \(p\).
Reviewer: Ananyo Dan (Sheffield)On the geometric P=W conjecturehttps://zbmath.org/1496.140332022-11-17T18:59:28.764376Z"Mauri, Mirko"https://zbmath.org/authors/?q=ai:mauri.mirko"Mazzon, Enrica"https://zbmath.org/authors/?q=ai:mazzon.enrica"Stevenson, Matthew"https://zbmath.org/authors/?q=ai:stevenson.matthewLet \(C\) be a Riemann surface of genus \(g\) and let \(G\) be a complex reductive algebraic group. If the character variety \(M_B=M_B(C,G)\) admits a compactification \(\bar M_B\) such that the pair \((\bar M_B,\bar M_B\setminus M_B)\) is a projective reduced dlt pair, then the geometric P=W conjecture is: the dual boundary complex of \(\bar M_B\setminus M_B\) is homotopy equivalent to a sphere. The geometric P=W conjecture is first formulated by \textit{L. Katzarkov} et al. [Commun. Math. Phys. 336, No. 2, 853--903 (2015; Zbl 1314.32021)]. In this paper, the authors confirmed the geometric P=W conjecture if \(r=1\), \(g\geq 1\) or \(r\geq 1\), \(g=1\). Indeed, they gave two proofs of the main results adopting non-archimedean geometry and degenerations of compact hyperkähler manifolds. On other hand, they also compared the geometric P=W conjecture with the cohomological P=W conjecture established by \textit{M. A. A. De Cataldo} et al. [Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)]. And, under some assumption, they proved that the geometric P=W conjecture implies the cohomological P=W conjecture at the highest weight. Finally, they also obtained some topological invariants of dual boundary complex for arbitrary genus, which can be viewed as partial evidence for the geometric P=W conjecture for general cases.
Reviewer: Yonghong Huang (Guangzhou)Poisson geometry of the moduli of local systems on smooth varietieshttps://zbmath.org/1496.140352022-11-17T18:59:28.764376Z"Pantev, Tony"https://zbmath.org/authors/?q=ai:pantev.tony-g"Toën, Bertrand"https://zbmath.org/authors/?q=ai:toen.bertrandThe authors examine the Poisson geometry of moduli spaces of local systems.
Let \(X\) be a smooth complex variety of dimension \(d\) and \(G\) a reductive group. If \(X\) is curve than it is well-known that the moduli space of \(G\)-local systems on \(X\) carries a canonical Poisson structure, whose symplectic leaves are moduli of \(G\)-local systems whose monodromy at infinity is fixed (up to conjugacy).
The authors give a natural extension of this result to the case of higher dimensions, however even the correct formulation now invariably involves derived geometry. If \(d > 1\) then the moduli space of \(G\)-local systems is naturally replaced by a derived moduli stack \(\mathrm{Loc}_G(X)\) of \(G\)-local systems (note that this takes into account the whole homotopy type of \(X\) as opposed to just the fundamental group). It follows from earlier results of the authors and their collaborators that for any compact oriented (real) manifold \(M\) there is a \((2 - \dim_{ \mathbb R} M)\)-shifted symplectic structure on \(\mathrm{Loc}_G(M)\) and thus a non-degenerate \((2 - \dim_{ \mathbb R} M)\)-shifted Poisson structure.
In this new paper the authors show that if \(X\) is a smooth complex variety, not necessarily proper, \(\mathrm{Loc}_G(X)\) carries a canonical \((2-2d)\)-shifted Poission structure.
They moreover describe some generalized symplectic leaves of the foliation if \(X\) admits a smooth compactification whose divisor at infinity is simple normal crossing with at most double intersections. Analogously to the case of curves these leaves are given by derived moduli of \(G\)-local system with fixed local monodromy at infinity as long as a technical condition the authors call `strictness' is satisfied.
As the authors say this is a first step towards understanding moduli of local systems on higher dimensional open varieties, with nonproper generalizations of Simpson's nonabelian Hodge theory as a key long term motivation.
The key ingredient in the proof is the restriction map to the boundary at infinity. Any smooth complex algebraic variety has well-defined boundary at infinity \(\partial X\) that is a compact manifold of real dimension \(2d-1\). Now \(\mathrm{Loc}_G(\partial G)\) has a shifted symplectic structure and by results of Calaque the restriction map is Lagrangian [\textit{D. Calaque}, Contemp. Math. 643, 1--23 (2015; Zbl 1349.14005)]. Then this induces the Poisson structure on \(\mathrm{Loc}_G(X)\) by \textit{V. Melani} and \textit{P. Safronov} [Sel. Math., New Ser. 24, No. 4, 3061--3118 (2018; Zbl 1461.14006); Sel. Math., New Ser. 24, No. 4, 3119--3173 (2018; Zbl 1440.14004)]. The characterization of symplectic leaves is more involved, already the notion of `fixing the monodromy at infinity' is more complicated than in the 1-dimensional case.
The paper includes useful discussions of algebraic descriptions of the moduli of \(G\)-local systems and of the boundary at infinity of a smooth complex variety. The special case that \(X\) is a curve is discussed in detail. The authors expect (but do not show) that in this case the 0-shifted symplectic structure they construct agrees with the one that is known from the literature.
Reviewer: Julian Holstein (Hamburg)On Ricci negative derivationshttps://zbmath.org/1496.170072022-11-17T18:59:28.764376Z"Gutiérrez, María Valeria"https://zbmath.org/authors/?q=ai:gutierrez.maria-valeriaThe present paper is a contribution to the problem of negative Ricci curvature in the homogeneous setting. A question considered is the following: Given a nilpotent Lie algebra \(\mathfrak{n}\), which are the solvable Lie algebras with nilradical \(\mathfrak{n}\) admitting a metric with \(\mathrm{Ric}<0\)? Lauret and Will have conjectured that given a nilpotent Lie algebra the space of all digonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature, coincides with an open convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. The author proves the validity of this conjecture in dimensions \(\le 5\), as well as for Heisenberg Lie algebras and standard filiform Lie algebras. A related conjecture has been also posed by Nikolayevsky and Nikonorov.
Reviewer: Andreas Arvanitoyeorgos (Patras)Vertex algebras associated with hypertoric varietieshttps://zbmath.org/1496.170212022-11-17T18:59:28.764376Z"Kuwabara, Toshiro"https://zbmath.org/authors/?q=ai:kuwabara.toshiroThe functors known as classical and quantum Hamiltonian reduction enjoy an algebraic realisation in terms of BRST cohomology and a geometric one building on nilpotent orbits, arc spaces and Slodowy slices. There is therefore a lot of interest in better understanding the relationships between these two approaches.
This article considers the quantum affine version of the reduction that produces hypertoric varieties. A quantisation of these varieties has already been constructed [\textit{I. M. Musson} and \textit{M. Van den Bergh}, Invariants under tori of rings of differential operators and related topics. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0928.16019)] and is known as a quantised hypertoric algebra. Here, an affinisation of this, meaning a vertex algebra, is proposed by carefully analysing the jet bundle over the hypertoric variety. It is moreover identified as a coset of a number of symplectic bosons (\(\beta\gamma\) systems) by a number of free bosons (Heisenberg vertex algebras). Finally, the Zhu algebra of this vertex algebra is shown to be at least a subalgebra of the corresponding quantised hypertoric algebra.
Heisenberg cosets of \(\beta\gamma\) systems have received a lot of attention in the literature, see for example [\textit{A. R. Linshaw}, J. Pure Appl. Algebra 213, No. 5, 632--648 (2009; Zbl 1230.17023)]. In the final section, this work is used to identify certain special cases of the proposed affinisation of the quantised hypertoric algebra. In particular, one particularly interesting special case turns out to give the subregular W-algebra of \(\mathfrak{sl}_n\) at non-admissible level \(-n+1\).
Reviewer: David Ridout (Melbourne)\(\mathfrak{L}\)-prolongations of graded Lie algebrashttps://zbmath.org/1496.170222022-11-17T18:59:28.764376Z"Marini, Stefano"https://zbmath.org/authors/?q=ai:marini.stefano"Medori, Costantino"https://zbmath.org/authors/?q=ai:medori.costantino"Nacinovich, Mauro"https://zbmath.org/authors/?q=ai:nacinovich.mauroAuthors' abstract: In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.
Reviewer: V. V. Gorbatsevich (Moskva)Local formulas for multiplicative formshttps://zbmath.org/1496.220012022-11-17T18:59:28.764376Z"Cabrera, A."https://zbmath.org/authors/?q=ai:cabrera.alejandro"Mărcuţ, I."https://zbmath.org/authors/?q=ai:marcut.ioan"Salazar, M. A."https://zbmath.org/authors/?q=ai:salazar.maria-ameliaIn a previous paper, the authors gave an explicit construction of a local Lie groupoid, for a given Lie algebroid, which is not necessarily integrable. In the paper under review, they extend this integration process to Lie algebroids endowed with extra infinitesimal data. This way, they give explicit formulas for the local version of various multiplicative forms on groupoids. Explicitly, they obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids.
Reviewer: Iakovos Androulidakis (Athína)Riemannian submersions of \(\mathrm{SO}_0(2,1)\)https://zbmath.org/1496.220132022-11-17T18:59:28.764376Z"Byun, Taechang"https://zbmath.org/authors/?q=ai:byun.taechang\textit{U. Pinkall} [Invent. Math. 81, 379--386 (1985; Zbl 0585.53051)] proved that, for the Hopf bundle \(S^1\to S^3\to S^2\), the parallel displacement along a simple closed curve in the base space depends only on the area surrounded by the curve. The paper under review studies similar question for the Riemannian submersions \(G \to N\backslash G\), \(G \to A\backslash G\), \(G \to K\backslash G\), and \(G \to NA\backslash G\), where \(G=NAK\) is the Iwasawa decomposition of the Lie group \(G=\mathrm{SO}^0(2,1)\).
Reviewer: Anton Galaev (Hradec Králové)Conformally formal manifolds and the uniformly quasiregular non-ellipticity of \((\mathbb{S}^2 \times\mathbb{S}^2)\#(\mathbb{S}^2\times\mathbb{S}^2)\)https://zbmath.org/1496.300152022-11-17T18:59:28.764376Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmariThe paper is devoted to quasiregular self-maps answering an open question. Given two oriented Riemann \(n\)-manifolds \(M\) and \(N\), a map \(f:M\to N\) is \(K\)-quasiregular, \(K\geq1\), if \(f\) is continuous, \(f\in W_{\text{loc}}^{1,n}(M,N)\) and \(|Df(x)|^n\leq KJ_f(x)\) for almost every \(x\in M\). Here, \(|\cdot|\) is the operator norm, and \(J_f\) is the Jacobian determinant. A \(K\)-quasiregular homeomorphism is called \(K\)-quasiconformal. A self-map \(M\to M\) is uniformly \(K\)-quasiregular if every iterate of \(f\) is \(K\)-quasiregular. A closed, connected, oriented Riemann \(n\)-manifold \(M\) is quasiregularly elliptic if there exists a non-constant quasiregular map \(f:\mathbb R^n\to M\). Similarly, \(M\) is uniformly quasiregularly elliptic if there exists a non-constant non-injective uniformly quasiregular self-map \(f:M\to M\). The author resolves negatively an open question in the following theorem.
Theorem 1.1. The manifold \((\mathbb S^2\times\mathbb S^2)\#(\mathbb S^2\times\mathbb S^2)\) is not uniformly quasiregularly elliptic.
The author gives also the concrete topological obstruction for uniformly quasiregular ellipticity.
Reviewer: Dmitri V. Prokhorov (Saratov)Deformations of balanced metricshttps://zbmath.org/1496.320132022-11-17T18:59:28.764376Z"Sferruzza, Tommaso"https://zbmath.org/authors/?q=ai:sferruzza.tommasoThe author studies small deformations of balanced metrics. As main result, he obtains a neccessary condition for the existence of a deformation of balanced metrics parameterized by a single complex variable, which is an equation on the central fiber, concerning \(\partial\) and \(\overline\partial\) operators.
Reviewer: Quanting Zhao (Wuhan)Short closed geodesics on cusped hyperbolic surfaceshttps://zbmath.org/1496.320142022-11-17T18:59:28.764376Z"Vo, Hanh"https://zbmath.org/authors/?q=ai:vo.hanh-nguyenSummary: This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any nonnegative integer \(k\), we consider the set of closed geodesics that self-intersect at least \(k\) times and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly \(k\) for large enough \(k\).A Schwarz lemma for weakly Kähler-Finsler manifoldshttps://zbmath.org/1496.320152022-11-17T18:59:28.764376Z"Nie, Jun"https://zbmath.org/authors/?q=ai:nie.jun"Zhong, Chunping"https://zbmath.org/authors/?q=ai:zhong.chunpingSummary: In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant.On Monge-Ampère volumes of direct imageshttps://zbmath.org/1496.320222022-11-17T18:59:28.764376Z"Finski, Siarhei"https://zbmath.org/authors/?q=ai:finski.siarheiSummary: This paper is devoted to the study of the asymptotics of Monge-Ampère volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of those admitting projectively flat Hermitian structures.Correction to: ``Logarithmic Bergman kernel and conditional expectation of Gaussian holomorphic fields''https://zbmath.org/1496.320252022-11-17T18:59:28.764376Z"Sun, Jingzhou"https://zbmath.org/authors/?q=ai:sun.jingzhouCorrection to the author's paper [ibid. 31, No. 8, 8520--8538 (2021; Zbl 1480.32005)].Totally geodesic discs in bounded symmetric domainshttps://zbmath.org/1496.320292022-11-17T18:59:28.764376Z"Kim, Sung-Yeon"https://zbmath.org/authors/?q=ai:kim.sung-yeon"Seo, Aeryeong"https://zbmath.org/authors/?q=ai:seo.aeryeongSummary: In this paper, we characterize \(C^2\)-smooth totally geodesic isometric embeddings \(f:\Omega \rightarrow \Omega^{\prime}\) between bounded symmetric domains \(\Omega\) and \(\Omega^{\prime}\) which extend \(C^1\)-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if \(\Omega\) is irreducible, there exist totally geodesic bounded symmetric subdomains \(\Omega_1\) and \(\Omega_2\) of \(\Omega^{\prime}\) such that \(f = (f_1, f_2)\) maps into \(\Omega_1\times \Omega_2\subset \Omega\) where \(f_1\) is holomorphic and \(f_2\) is anti-holomorphic totally geodesic isometric embeddings. If \(\mathrm{rank}(\Omega^{\prime})<2\mathrm{rank}(\Omega)\), then either \(f\) or \({\bar{f}}\) is a standard holomorphic embedding.A Cartan-Hartogs version of the polydisk theoremhttps://zbmath.org/1496.320332022-11-17T18:59:28.764376Z"Mossa, Roberto"https://zbmath.org/authors/?q=ai:mossa.roberto"Zedda, Michela"https://zbmath.org/authors/?q=ai:zedda.michelaSummary: We extend the Polydisk Theorem for symmetric bounded domains to Cartan-Hartogs domains, and apply it to prove that a Cartan-Hartogs domain inherits totally geodesic submanifolds from the bounded symmetric domain which is based on, and to give a characterization of Cartan-Hartogs's geodesics with linear support.On a \(k\)-th Gauduchon almost Hermitian manifoldhttps://zbmath.org/1496.320362022-11-17T18:59:28.764376Z"Kawamura, Masaya"https://zbmath.org/authors/?q=ai:kawamura.masayaSummary: We characterize the \(k\)-th Gauduchon condition and by applying its characterization, we reprove that a compact \(k\)-th Gauduchon, semi-Kähler manifold becomes quasi-Kähler, which tells us that in particular, a compact almost pluriclosed, semi-Kähler manifold is quasi-Kähler.Compact almost Hermitian manifolds with quasi-negative curvature and the almost Hermitian curvature flowhttps://zbmath.org/1496.320372022-11-17T18:59:28.764376Z"Kawamura, Masaya"https://zbmath.org/authors/?q=ai:kawamura.masayaSummary: We show that along the almost Hermitian curvature flow, the non-positivity of the first Chern-Ricci curvature can be preserved if the initial almost Hermitian metric has the Griffiths non-positive Chern curvature. If additionally, the first Chern-Ricci curvature of the initial metric is negative at some point, then we show that the almost complex structure of a compact non-quasi-Kähler almost Hermitian manifold equipped with such a metric cannot be integrable.An a priori \(C^0\)-estimate for the Fu-Yau equation on compact almost astheno-Kähler manifoldshttps://zbmath.org/1496.320382022-11-17T18:59:28.764376Z"Kawamura, Masaya"https://zbmath.org/authors/?q=ai:kawamura.masayaSummary: We investigate the Fu-Yau equation on compact almost astheno-Kähler manifolds and show an \textit{a priori} \(C^0\)-estiamte for a smooth solution of the equation.On an a priori \(L^\infty\) estimate for a class of Monge-Ampère type equations on compact almost Hermitian manifoldshttps://zbmath.org/1496.320392022-11-17T18:59:28.764376Z"Kawamura, Masaya"https://zbmath.org/authors/?q=ai:kawamura.masayaSummary: We investigate Monge-Ampère type equations on almost Hermitian manifolds and show an \textit{a priori} \(L^\infty\) estimate for a smooth solution of these equations.On the dimension of Dolbeault harmonic \((1,1)\)-forms on almost Hermitian \(4\)-manifoldshttps://zbmath.org/1496.320412022-11-17T18:59:28.764376Z"Piovani, Riccardo"https://zbmath.org/authors/?q=ai:piovani.riccardo"Tomassini, Adriano"https://zbmath.org/authors/?q=ai:tomassini.adrianoSummary: We prove that the dimension \(h^{1,1}_{\overline{\partial}}\) of the space of Dolbeault harmonic \((1,1)\)-forms is not necessarily always equal to \(b^-\) on a compact almost complex \(4\)-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact \(4\)-manifolds with \(h^{1,1}_{\overline{\partial}}=b^-+1\). This gives an answer to [Math. Z. 302, No. 1, 47--72 (2022; Zbl 1496.32035)] by \textit{T. Holt}.Global surfaces of Section with positive genus for dynamically convex Reeb flowshttps://zbmath.org/1496.320422022-11-17T18:59:28.764376Z"Hryniewicz, Umberto L."https://zbmath.org/authors/?q=ai:hryniewicz.umberto-l"Salomão, Pedro A. S."https://zbmath.org/authors/?q=ai:salomao.pedro-a-s"Siefring, Richard"https://zbmath.org/authors/?q=ai:siefring.richardSummary: We establish some new existence results for global surfaces of section of dynamically convex Reeb flows on the three-sphere. These sections often have genus, and are the result of a combination of pseudoholomorphic methods with some elementary ergodic methods.Super \(J\)-holomorphic curves: construction of the moduli spacehttps://zbmath.org/1496.320432022-11-17T18:59:28.764376Z"Keßler, Enno"https://zbmath.org/authors/?q=ai:kessler.enno"Sheshmani, Artan"https://zbmath.org/authors/?q=ai:sheshmani.artan"Yau, Shing-Tung"https://zbmath.org/authors/?q=ai:yau.shing-tungSummary: Let \(M\) be a super Riemann surface with holomorphic distribution \({\mathcal{D}}\) and \(N\) a symplectic manifold with compatible almost complex structure \(J\). We call a map \(\Phi :M\rightarrow N\) a super \(J\)-holomorphic curve if its differential maps the almost complex structure on \({\mathcal{D}}\) to \(J\). Such a super \(J\)-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super \(J\)-holomorphic curves as a smooth subsupermanifold of the space of maps \(M\rightarrow N\).On the definition of irreducible holomorphic symplectic manifolds and their singular analogshttps://zbmath.org/1496.320442022-11-17T18:59:28.764376Z"Schwald, Martin"https://zbmath.org/authors/?q=ai:schwald.martinSummary: In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss holomorphic symplectic, finite quotients of complex tori with \(\mathrm{h}^0(X,\,\Omega^{[2]}_X)=1\) and their Lagrangian fibrations. Neither \(X\) nor the base can be smooth unless \(X\) is a \(2\)-torus.The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flowshttps://zbmath.org/1496.320602022-11-17T18:59:28.764376Z"Guedj, Vincent"https://zbmath.org/authors/?q=ai:guedj.vincent"Lu, Chinh H."https://zbmath.org/authors/?q=ai:lu.chinh-h"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedLet \(T>0\), \(\Omega\) be a bounded strictly pseudoconvex domain in \(\mathbb C^n\) and \(\Omega_T=(0,T)\times\Omega\). In this paper, the authors consider the following degenerate complex Monge-Ampère flow (CMAF), on \(\Omega_T\):
\[
dt\wedge(dd^cu)^n=e^{\partial_tu+F(t,z,u)}g(z)dt\wedge dV,
\]
where \(dV\) is the Euclidean volume on \(\mathbb C^n\), \(g\in L^p(\Omega)\) for some \(p>1\) and \(g>0\) a.e., \(F(t,z,r)\) is continuous on \([0,T)\times\Omega\times\mathbb R\) and bounded on \([0,T)\times\Omega\times J\) for each compact \(J\subset\mathbb R\), \(F\) is increasing in \(r\) and \((t,r)\to F(t,\cdot,r)\) is uniformly Lipschitz and semi-convex. This can be seen as a local version of the pluripotential Kähler-Ricci flow studied by the authors in the earlier work [Geom. Topol. 24, No. 3, 1225--1296 (2020; Zbl 1458.32035)].
To deal with the above equation (CMAF), the authors introduce and study the family \(\mathcal P(\Omega_T)\) of parabolic potentials. These are functions \(u:\Omega_T\to\mathbb R\cup\{-\infty\}\) whose slices \(u(t,\cdot)\) are plurisubharmonic and such that the family \(\{u(\cdot,z):\,z\in\Omega\}\) is locally uniformly Lipschitz in \((0,T)\). They prove in particular that the parabolic complex Monge-Ampère operator \(dt\wedge(dd^cu)^n\) is well defined as a measure for \(u\in\mathcal P(\Omega_T)\cap L^\infty_{\mathrm{loc}}(\Omega_T)\) and they introduce the notions of sub/super/solution to (CMAF) in this class. They also prove a comparison theorem for bounded parabolic potentials.
A function \(h\) defined on the parabolic boundary \(\big([0,T)\times\partial\Omega\big)\cup(\{0\}\times\Omega)\) of \(\Omega_T\) is a Cauchy-Dirichlet boundary data if \(h\) is continuous on \([0,T)\times\partial\Omega\), the family \(\{h(\cdot,z):\,z\in\partial\Omega\}\) is locally uniformly Lipschitz in \((0,T)\), \(h(0,\cdot)\in \operatorname{PSH}(\Omega)\cap L^\infty(\Omega)\) and \(\lim_{\Omega\ni z\to\zeta}h(0,z)=h(0,\zeta)\) for all \(\zeta\in\partial\Omega\).
The main result of the paper is that the Cauchy-Dirichlet problem for (CMAF) can be solved by the Perron method with parabolic potentials as admissible functions. Assuming that the Cauchy-Dirichlet boundary data \(h\) is such that for every \(S\in(0,T)\) there exists a constant \(C(S)\) with
\[
t|\partial_th(t,z)|\leq C(S) \text{ and } t^2\partial_t^2h(t,z)\leq C(S),\;\forall\,(t,z)\in (0,S]\times\partial\Omega,
\]
the authors prove that the pluripotential solution to this Cauchy-Dirichlet problem is given by the upper envelope of the family of all subsolutions with boundary data \(h\).
Reviewer: Dan Coman (Syracuse)Planar immersions with prescribed curl and Jacobian determinant are uniquehttps://zbmath.org/1496.350112022-11-17T18:59:28.764376Z"Gruber, Anthony"https://zbmath.org/authors/?q=ai:gruber.anthonySummary: We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.Convergence of the weighted Yamabe flowhttps://zbmath.org/1496.350942022-11-17T18:59:28.764376Z"Yan, Zetian"https://zbmath.org/authors/?q=ai:yan.zetianSummary: We introduce the weighted Yamabe flow
\[
\begin{cases}
\frac{\partial g}{\partial t} = (r_{\phi}^m - R_{\phi}^m) g \\
\frac{\partial \phi}{\partial t} = \frac{m}{2} (R_{\phi}^m - r_{\phi}^m)
\end{cases}
\]
on a smooth metric measure space \((M^n, g, e^{-\phi} \operatorname{dvol}_g, m)\), where \(R_{\phi}^m\) denotes the associated weighted scalar curvature, and \(r_{\phi}^m\) denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension \(n\) satisfies \(n \geqslant 3\).Boundary regularity of minimal oriented hypersurfaces on a manifoldhttps://zbmath.org/1496.351352022-11-17T18:59:28.764376Z"Steinbrüchel, Simone"https://zbmath.org/authors/?q=ai:steinbruchel.simoneSummary: In this article we prove that all boundary points of a minimal oriented hypersurface in a Riemannian manifold are regular, that is, in a neighborhood of any boundary point, the minimal surface is a \(C^{1, \frac{1}{4}}\) submanifold with boundary.An \(\varepsilon\)-regularity result with mean curvature control for Willmore immersions and application to minimal bubbling.https://zbmath.org/1496.352122022-11-17T18:59:28.764376Z"Marque, Nicolas"https://zbmath.org/authors/?q=ai:marque.nicolasSummary: In this paper, we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end, we replace the total curvature control in the proof of the \(\varepsilon\)-regularity for Willmore immersions by a control of the local Willmore energy.On an asymptotically log-periodic solution to the graphical curve shortening flow equationhttps://zbmath.org/1496.352372022-11-17T18:59:28.764376Z"Tsai, Dong-Ho"https://zbmath.org/authors/?q=ai:tsai.dong-ho"Wang, Xiao-Liu"https://zbmath.org/authors/?q=ai:wang.xiaoliuSummary: With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution \(y\left(x, t\right) \) has the interesting property that it converges to a log-periodic function of the form \[A\sin \left( \log t\right) +B\cos \left( \log t\right)\] as \(t\rightarrow \infty\), where \(A, B\) are constants. Moreover, for any two numbers \(\alpha <\beta,\) we are also able to construct a solution satisfying the oscillation limits \[\liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha, \quad \limsup\limits_{t\rightarrow \infty}y\left( x,t\right) = \beta, \quad x\in K \] on any compact subset \(K\subset \mathbb{R} \).Existence of hypercylinder expanders of the inverse mean curvature flowhttps://zbmath.org/1496.352402022-11-17T18:59:28.764376Z"Hui, Kin Ming"https://zbmath.org/authors/?q=ai:hui.kin-mingSummary: We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in \(\mathbb{R}^n\times \mathbb{R}\), \(n\ge 2\), of the form \((r,y(r))\) or \((r(y),y)\), where \(r=|x|\), \(x\in \mathbb{R}^n\), is the radially symmetric coordinate and \(y\in \mathbb{R}\). More precisely, for any \(\lambda>\frac{1}{n-1}\) and \(\mu>0\), we will give a new proof of the existence of a unique even solution \(r(y)\) of the equation \(\frac{r^{\prime \prime}(y)}{1+r^{\prime}(y)^2}=\frac{n-1}{r(y)}-\frac{1+r^{\prime}(y)^2}{\lambda (r(y)-yr^{\prime}(y))}\) in \(\mathbb{R}\) which satisfies \(r(0)=\mu\), \(r^{\prime}(0)=0\) and \(r(y)>yr^{\prime}(y)>0\) for any \(y\in \mathbb{R}\). We will prove that \(\lim_{y\to \infty}r(y)=\infty\) and \(a_1:=\lim_{y\to \infty}r^{\prime}(y)\) exists with \(0\le a_1<\infty\). We will also give a new proof of the existence of a constant \(y_1>0\) such that \(r^{\prime \prime}(y_1)=0\), \(r^{\prime \prime}(y)>0\) for any \(0<y<y_1\), and \(r^{\prime \prime}(y)<0\) for any \(y>y_1\) .On the first eigenvalue of the Laplacian on compact surfaces of genus threehttps://zbmath.org/1496.352682022-11-17T18:59:28.764376Z"Ros, Antonio"https://zbmath.org/authors/?q=ai:ros.antonioSummary: For any compact Riemannian surface of genus three \((\Sigma, ds^2)\) Yang and Yau proved that the product of the first eigenvalue of the Laplacian \(\lambda_1(ds^2)\) and the area \(\mathit{Area}(ds^2)\) is bounded above by \(24\pi\). In this paper we improve the result and we show that \(\lambda_1(ds^2) \mathit{Area}(ds^2) \leq 16(4 - \sqrt{7})\pi \approx 21.668\pi\). About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value \(\approx 21.414\pi\).Weak-strong uniqueness for the Navier-Stokes equation for two fluids with ninety degree contact angle and same viscositieshttps://zbmath.org/1496.352792022-11-17T18:59:28.764376Z"Hensel, Sebastian"https://zbmath.org/authors/?q=ai:hensel.sebastian"Marveggio, Alice"https://zbmath.org/authors/?q=ai:marveggio.aliceSummary: We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier-Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. In the present work, we devise for the resulting evolution problem a suitable weak solution concept based on the framework of varifolds and establish as the main result a weak-strong uniqueness principle in 2D. The proof is based on a relative entropy argument and requires a non-trivial further development of ideas from the recent work of \textit{J. Fischer} and the first author [Arch. Ration. Mech. Anal. 236, No. 2, 967--1087 (2020; Zbl 1465.35332)] to incorporate the contact angle condition. To focus on the effects of the necessarily singular geometry of the evolving fluid domains, we work for simplicity in the regime of same viscosities for the two fluids.Nonlocal diffusion of smooth setshttps://zbmath.org/1496.354172022-11-17T18:59:28.764376Z"Attiogbe, Anoumou"https://zbmath.org/authors/?q=ai:attiogbe.anoumou"Fall, Mouhamed Moustapha"https://zbmath.org/authors/?q=ai:fall.mouhamed-moustapha"Thiam, El Hadji Abdoulaye"https://zbmath.org/authors/?q=ai:thiam.el-hadji-abdoulayeSummary: We consider normal velocity of smooth sets evolving by the \(s\)-fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for \(s\in [\frac{1}{2}, 1)\) while, for \(s\in (0, \frac{1}{2}) \), it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.Heat flow for harmonic maps from graphs into Riemannian manifoldshttps://zbmath.org/1496.370382022-11-17T18:59:28.764376Z"Baird, Paul"https://zbmath.org/authors/?q=ai:baird.paul"Fardoun, Ali"https://zbmath.org/authors/?q=ai:fardoun.ali"Regbaoui, Rachid"https://zbmath.org/authors/?q=ai:regbaoui.rachidSummary: We introduce the notion of harmonic map from a graph into a Riemannian manifold via a discrete version of the energy density. Existence and basic properties are established. Global existence and convergence of the associated heat flow are proved without any assumption on the curvature of the target manifold. We discuss a variant of the Steiner problem which replaces length by elastic energy.A generalized Poincaré-Birkhoff theoremhttps://zbmath.org/1496.370692022-11-17T18:59:28.764376Z"Moreno, Agustin"https://zbmath.org/authors/?q=ai:moreno.agustin-s"van Koert, Otto"https://zbmath.org/authors/?q=ai:van-koert.ottoSummary: We prove a generalization of the classical Poincaré-Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem [\textit{A. Moreno} and \textit{O. van Koert}, Nonlinearity 35, No. 6, 2920--2970 (2022; Zbl 07539293)].Contact Hamiltonian and Lagrangian systems with nonholonomic constraintshttps://zbmath.org/1496.370712022-11-17T18:59:28.764376Z"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuel"Jiménez, Víctor M."https://zbmath.org/authors/?q=ai:jimenez.victor-manuel"Lainz, Manuel"https://zbmath.org/authors/?q=ai:lainz.manuelThis paper aims at using contact and Jacobi geometry to develop the natural geometric framework for studying the dynamics of mechanical systems that are subject to both nonholonomic constraints and Rayleigh dissipation.
A \textit{nonholonomic mechanical system} is a mechanical system subject to \textit{nonholonomic constraints}, i.e., constraints (on the position and velocities) that do not derive from constraints only on the positions. Examples include mechanical systems that have rolling contact (like a ball rolling without slipping on a plane) or some kind of sliding contact (like a rigid body sliding on a plane). In the Lagrangian formalism, a mechanical system is described by a \textit{Lagrangian function} \(L\colon TQ\to\mathbb{R},\ (q,\dot q)\mapsto L(q,\dot q)\), where the smooth manifold \(Q\) denotes the \textit{configuration space} of the system. Then a nonholonomic constraint is given by a submanifold \(\mathcal{D}\subset TQ\) such that \(\tau_Q(\mathcal{D})=Q\), where \(\tau_Q\colon TQ\to Q\) denotes the bundle map. In the following, one only considers nonholonomic constraints that are linear in the velocities, i.e., \(\mathcal{D}\subset TQ\) is a vector subbundle.
If the mechanical system is conservative, i.e., \(L=K_g-V\), where \(V\in C^\infty(Q)\) and \(K_g\) is the kinetic energy of some pseudo-Riemannian metric \(g\) on \(Q\), then the Lagrangian \(L\) is regular, i.e., the associated Legendre transform \(\mathbb{F}L:TQ\to T^\ast Q\) is a local diffeomorphism. In this case, the natural geometric description of their dynamics is provided in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{R. Abraham} and \textit{J. E. Marsden}, Foundations of mechanics. 2nd ed., rev., enl., and reset. With the assistance of Tudor Ratiu and Richard Cushman. Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc (1978; Zbl 0393.70001)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ,\omega_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\omega_L=(\mathbb{F}L)^\ast\omega_{\text{can}}\) is the pull-back along \(\mathbb{F}L\) of the canonical symplectic form on \(T^\ast Q\). This \(\Gamma_L\) is a SODE (second-order differential equation) on \(TQ\) and its flow is obtained integrating the standard Euler-Lagrange equations. This description of the dynamics is consistent with the one arising from D'Alembert principle.
If the conservative mechanical system is additionally subject to nonholonomic constraints, then its dynamics can be still described in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{C.-M. Marle}, Rep. Math. Phys. 42, No. 1--2, 211--229 (1998; Zbl 0931.37023)] and references therein). Indeed, its dynamics is the projection on \(Q\) of the flow of a nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \((TTQ)|_\mathcal{D}\). This description of the nonholonomic dynamics is consistent with the one arising from Chetaev version of D'Alembert principle. Moreover, this nonholonomic dynamics is almost-Poisson but not Poisson. Indeed, there is a bracket \(\{-,-\}\) on \(C^\infty(\mathcal{D})\) that satisfies the Leibniz rule in each entry and, together with the energy \(E_L\) on \(TQ\), controls the time evolution of the observables, but in generally it fails to satisfy the Jacobi identity.
The authors start from the observation that there are other kinds of nonholonomic mechanical systems that do not fit in the previous framework. As a first example, one can consider a nonholonomic mechanical system that is also subject to Rayleigh dissipation and so non-conservative. Additional examples come from thermodynamics. These mechanical systems can be described by a Lagrangian function \(L\colon TQ\times\mathbb{R}\to\mathbb{R},\ (q,\dot q,z)\mapsto L(q,\dot q,z):=L_z(q,\dot q),\) where the smooth manifold \(Q\) is the configuration space and the parameter \(z\) on \(\mathbb{R}\) denotes friction (or a thermal variable in thermodynamics).
If the Lagrangian \(L\) is regular, in the sense that, for any \(z\in\mathbb{R}\), the associated Legendre transform \(\mathbb{F}L_z:TQ\to T^\ast Q\) is a local diffeomorphism, the natural geometric framework of their dynamics is provided by the theory of Hamiltonian systems on contact manifolds (cf., e.g., [\textit{M. de León} and \textit{M. Lainz Valcázar}, J. Math. Phys. 60, No. 10, 102902, 18 p. (2019; Zbl 1427.70039)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ\times\mathbb{R},\eta_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\eta_L=(\mathbb{F}L\times\operatorname{id}_\mathbb{R})^\ast\eta_{\text{can}}\) is the pull-back along \(\mathbb{F}L\times\operatorname{id}_\mathbb{R}\) of the canonical contact form on \(T^\ast Q\times\mathbb{R}\). In the current setting, this \(\Gamma_L\) is still an SODE on \(TQ\times\mathbb{R}\) (in the sense recalled in Definition~5) and its flow is obtained integrating the so-called Herglotz equations. Indeed, this description of the dynamics is consistent with the one arising from the Herglotz variational principle (as recalled in Section~4).
In this paper the authors show that the theory of Hamiltonian systems on contact manifolds can be adapted to provide a geometric interpretation of the dynamics of mechanical systems that are subject to both dissipation and nonholonomic constraints. Section~5 defines a version of the Herglotz principle in presence of nonholonomic constraints: essentially, one restricts the variations so that they satisfy the constraints. Then the dynamics is described by the extremals of this Herglotz principle with constraints and they are given by the solutions of the so-called constrained Herglotz equations (see Theorem~5). Actually, this description admits a geometric description similar to the one obtained when there are no constraints. Indeed, Theorem 6 shows that, if the Lagrangian \(L\colon TQ\times\mathbb{R}\to\mathbb{R}\) is regular (i.e., \(F_z\colon TQ\to\mathbb{R}\) is regular, for any \(z\in\mathbb{R}\), as recalled above), the solutions of the constrained Herglotz equations are the projections on \(Q\) of the integral curves of the nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\times\mathbb{R}\) (see Definition~5) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \(T(TQ\times\mathbb{R})\) along \(\mathcal{D}\).
The authors also prove that the time evolution of these mechanical systems subject to both dissipation and nonholonomic constraints is governed by an almost-Jacobi bracket (see Definition 7). Indeed, in Section 6, they first construct a nonholonomic bracket from functions on \(TQ\times\mathbb{R}\) to functions on \(\mathcal{D}\times\mathbb{R}\) (see Equation 100), then they prove that this nonholonomic bracket (together with the Energy \(E_L\)) controls the time evolution of the observables (see Theorem 12) and it is an almost Jacobi bracket (see Proposition 6). Further, it turns out that this nonholonomic bracket is actually a Jacobi structure (i.e., it satisfies the Jacobi identity) if and only if the constraint \(\mathcal{D}\subset TQ\) is an involutive vector subbundle (see Theorem 13).
Finally, in Example 2, the authors illustrate their results applying them to a particular example given by a model of the Chaplygin's sleight subject to Rayleigh dissipation.
Reviewer: Alfonso Giuseppe Tortorella (Porto)On BMO and Carleson measures on Riemannian manifoldshttps://zbmath.org/1496.420292022-11-17T18:59:28.764376Z"Brazke, Denis"https://zbmath.org/authors/?q=ai:brazke.denis"Schikorra, Armin"https://zbmath.org/authors/?q=ai:schikorra.armin"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannickSummary: Let \(\mathcal{M}\) be a Riemannian \(n\)-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions \(u: \mathcal{M}\to\mathbb{R}\) by a Carleson measure condition of their \(\sigma\)-harmonic extension \(U: \mathcal{M}\times (0,\infty)\to\mathbb{R}\). We make crucial use of a \(T(b)\) theorem proved by \textit{S. Hofmann} et al. [\(L^p\)-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1371.28004)]. As an application, we show that the famous theorem of Coifman-Lions-Meyer-Semmes [\textit{R. Coifman} et al., J. Math. Pures Appl. (9) 72, No. 3, 247--286 (1993; Zbl 0864.42009)] holds in this class of manifolds: Jacobians of \(W^{1,n}\)-maps from \(\mathcal{M}\) to \(\mathbb{R}^n\) can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by \textit{E. Lenzmann} and \textit{A. Schikorra} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 193, Article ID 111375, 37 p. (2020; Zbl 1436.35012)] using only harmonic extensions, integration by parts, and trace space characterizations.Toric symplectic geometry and full spark frameshttps://zbmath.org/1496.420412022-11-17T18:59:28.764376Z"Needham, Tom"https://zbmath.org/authors/?q=ai:needham.tom"Shonkwiler, Clayton"https://zbmath.org/authors/?q=ai:shonkwiler.claytonSummary: The collection of \(d \times N\) complex matrices with prescribed column norms and singular values forms an algebraic variety, which we refer to as a frame space. Elements of frame spaces -- i.e., frames -- are used to give robust signal representations, so that geometrical properties of frame spaces are of interest to the signal processing community. This paper is concerned with the question: what is the probability that a frame drawn at random from a given frame space has the property that any subset of \(d\) of its columns gives a basis for \(\mathbb{C}^d\)? We show that the probability is one, generalizing recent work of \textit{J. Cahill} et al. [SIAM J. Appl. Algebra Geom. 1, No. 1, 38--72 (2017; Zbl 1370.42023)]. To prove this, we show that frame spaces are related to highly structured objects called toric symplectic manifolds. As another application, we characterize the norm and spectral data for which the corresponding frame space has singularities, answering an open question in the literature.Free metaplectic wavelet transform in \(L^2(\mathbb{R}^n)\)https://zbmath.org/1496.420452022-11-17T18:59:28.764376Z"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmad"Qadri, Huzaifa L."https://zbmath.org/authors/?q=ai:qadri.huzaifa-l"Lone, Waseem Z."https://zbmath.org/authors/?q=ai:lone.waseem-zThis paper introduces the notion of free-metaplectic wavelet transform in \(L^2(\mathbb{R}^n)\). The paper studies some fundamental properties such as the orthogonality relation, inversion formula, characterization of range, and the homogeneous approximation property of the proposed transform. Heisenberg and Pitt's uncertainty inequalities associated with the free-metaplectic wavelet transform are established. The analysis of the double-window wavelet transform in the free metaplectic domains is done. The uncertainty principles associated with the free metaplectic wavelet transform are given. It is shown that the transform provides an efficient time-frequency representation of non-transient multidimensional signals.
Reviewer: Yilun Shang (Newcastle)Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaceshttps://zbmath.org/1496.460112022-11-17T18:59:28.764376Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paulThe author furnishes an enhanced bound on the filling areas curves (not closed geodesics) in Banach spaces. He shows rigidity of \textit{P. M. Pu}'s classical systolic inequality [Pac. J. Math. 2, 55--71 (1952; Zbl 0046.39902)] and examines the isoperimetric constants of normed spaces.
Reviewer: Mohammed El Aïdi (Bogotá)The logarithmic Sobolev inequality for a submanifold in Euclidean spacehttps://zbmath.org/1496.460292022-11-17T18:59:28.764376Z"Brendle, Simon"https://zbmath.org/authors/?q=ai:brendle.simon.1Summary: We prove a sharp logarithmic Sobolev inequality that holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.Isoperimetric clusters in homogeneous spaces via concentration compactnesshttps://zbmath.org/1496.490232022-11-17T18:59:28.764376Z"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Paolini, Emanuele"https://zbmath.org/authors/?q=ai:paolini.emanuele"Stepanov, Eugene"https://zbmath.org/authors/?q=ai:stepanov.eugene"Tortorelli, Vincenzo Maria"https://zbmath.org/authors/?q=ai:tortorelli.vincenzo-mariaSummary: We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural ``relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being minimal among all generalized clusters with the same masses. The examples include any anisotropic perimeter in a Euclidean space, as well as a hyperbolic plane with the Riemannian perimeter and Heisenberg groups with a canonical left invariant perimeter or its equivalent versions.Variational approach to regularity of optimal transport maps: general cost functionshttps://zbmath.org/1496.490252022-11-17T18:59:28.764376Z"Otto, Felix"https://zbmath.org/authors/?q=ai:otto.felix"Prod'homme, Maxime"https://zbmath.org/authors/?q=ai:prodhomme.maxime"Ried, Tobias"https://zbmath.org/authors/?q=ai:ried.tobiasThe paper continues the line of research started with [\textit{M. Goldman} and \textit{F. Otto}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 5, 1209--1233 (2020; Zbl 1465.35263)] and [\textit{M. Goldman} et al., Commun. Pure Appl. Math. 74, No. 12, 2483--2560 (2021; Zbl 1480.35082)]. The aim is the study of the regularity theory for optimal transport maps with a purely variational approach.
Here the authors deal with general cost functions and Hölder continuous densities, obtaining a slightly more quantitative result than the one in the celebrated paper [\textit{G. De Philippis} and \textit{A. Figalli}, Publ. Math., Inst. Hautes Étud. Sci. 121, 81--112 (2015; Zbl 1325.49051)]. Besides the differences in the statement of the main result compared to the one of De Philippis and Figalli, the interest is in the different approach. Indeed, the authors do not rely on the regularity theory for the Monge-Ampère equation and use arguments similar to De Giorgi's strategy for the \(\varepsilon\)-regularity of minimal surfaces.
The result can be also applied to the study of the optimal transport problem on Riemannian manifolds with cost given by the square of the Riemannian distance.
Reviewer: Nicolò De Ponti (Trieste)Tensor trigonometry. Theory and applicationshttps://zbmath.org/1496.510012022-11-17T18:59:28.764376Z"Ninul, Anatoliĭ Sergeevich"https://zbmath.org/authors/?q=ai:ninul.anatolii-sergeevichFor the English edition see [Zbl 1482.51002].Volume properties and some characterizations of ellipsoids in \(\mathbb{E}^{n+1}\)https://zbmath.org/1496.520082022-11-17T18:59:28.764376Z"Kim, Dong-Soo"https://zbmath.org/authors/?q=ai:kim.dongsoo-s"Kim, Incheon"https://zbmath.org/authors/?q=ai:kim.incheon"Kim, Young Ho"https://zbmath.org/authors/?q=ai:kim.younghoSummary: Suppose that \(M\) is a strictly convex and closed hypersurface in \(\mathbb{E}^{n+1}\) with the origin \(o\) in its interior. We consider the homogeneous function \(g\) of positive degree \(d\) satisfying \(M=g^{-1}(1)\). Then, for a positive number \(h\) the level hypersurface \(g^{-1}(h)\) of \(g\) is a homothetic hypersurface of \(M\) with respect to the origin \(o\). In this paper, for tangent hyperplanes \(\Phi_h\) to \(g^{-1}(h)\) (\(0 < h < 1\)), we study the \((n + 1)\)-dimensional volume of the region enclosed by \(\Phi_h\) and the hypersurface \(M\), etc. As a result, with the aid of the theorem of Blaschke and Deicke for proper affine hypersphere centered at the origin, we establish some characterizations for ellipsoids in \(\mathbb{E}^{n+1}\). As a corollary, we extend Schneider's characterization for ellipsoids in \(\mathbb{E}^3\). Finally, for further study, we raise a question for elliptic paraboloids which was originally conjectured by Golomb.Properties of a curve whose convex hull covers a given convex bodyhttps://zbmath.org/1496.520122022-11-17T18:59:28.764376Z"Nikonorov, Yurii G."https://zbmath.org/authors/?q=ai:nikonorov.yurii-gSummary: In this note, we prove the following inequality for the norm \(N(K)\) of a convex body \(K\) in \(\mathbb{R}^n, n\geq 2\):
\[
N(K) \leq \frac{\pi^{\frac{n-1}{2}}}{2 \Gamma \left( \frac{n+1}{2}\right)}\cdot\,\text{length}\,(\gamma )+\frac{\pi^{\frac{n}{2}-1}}{\Gamma \left( \frac{n}{2}\right)} \cdot\,\text{diam}\,(K),
\]
where \(\text{diam}(K)\) is the diameter of \(K, \gamma\) is any curve in \(\mathbb{R}^n\) whose convex hull covers \(K\), and \(\Gamma\) is the gamma function. If in addition \(K\) has constant width \(\Theta\), then we get the inequality
\[
\text{length}\,(\gamma ) \geq \frac{2(\pi -1)\Gamma \left( \frac{n+1}{2}\right)}{\sqrt{\pi}\,\Gamma \left( \frac{n}{2}\right)}\cdot \Theta \geq 2(\pi -1) \cdot \sqrt{\frac{n-1}{2\pi}}\cdot \Theta.
\]
In addition, we pose several unsolved problems.Packings with geodesic and translation balls and their visualizations in \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) spacehttps://zbmath.org/1496.520202022-11-17T18:59:28.764376Z"Molnár, Emil"https://zbmath.org/authors/?q=ai:molnar.emil"Szirmai, Jenő"https://zbmath.org/authors/?q=ai:szirmai.jenoSummary: Remembering on our friendly cooperation between the Geometry Departments of Technical Universities of Budapest and Vienna (also under different names) a nice topic comes into consideration: the ``Gum fibre model'' (see Fig. 1).
One point of view is the so-called kinematic geometry by Vienna colleagues, e.g., as in [\textit{H. Stachel}, Math. Appl., Springer 581, 209--225 (2006; Zbl 1100.52005)], but also in very general context. The other point is the so-called \(\mathbf{H}^2 \times \mathbf{R}\) geometry and \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) geometry where -- roughly -- two hyperbolic planes as circle discs are connected with gum fibres, first: in a simple way, second: in a twisted way. This second homogeneous (Thurston) geometry will be our topic (initiated by Budapest colleagues, and discussed also in international cooperations).
We use for the computation and visualization of \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) its projective model, as in our previous papers. We found seemingly extremal geodesic ball packing for \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 9\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density
\(\approx 0.787758\) (Table 2). Much better translation ball packing is for group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 11\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density \(\approx 0.845306\) (Table 3).Topology of symplectomorphism groups and ball-swappingshttps://zbmath.org/1496.530012022-11-17T18:59:28.764376Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.2|li.jun.8|li.jun.3|li.jun.16|li.jun.1|li.jun.7|li.jun.6|li.jun|li.jun.11|li.jun.10"Wu, Weiwei"https://zbmath.org/authors/?q=ai:wu.weiwei|wu.weiwei.1|wu.weiwei.2Summary: In this survey article, we summarize some recent progress and problems on the symplectomorphism groups, with an emphasis on the connection to the space of ball-packings.
For the entire collection see [Zbl 1454.00057].On the first eigenvalue of the \(p\)-Laplacian on Riemannian manifoldshttps://zbmath.org/1496.530022022-11-17T18:59:28.764376Z"Seto, Shoo"https://zbmath.org/authors/?q=ai:seto.shooSummary: We survey results on the first (nontrivial) eigenvalue of the \(p\)-Laplace operator for both the Dirichlet and Neumann/closed condition on Riemannian manifolds. We also discuss an extension of the \(p\)-Laplace operator to act on differential forms. Some potential future directions of work are also given.
For the entire collection see [Zbl 1495.53004].Hamiltonian and Lagrangian systems in contact geometryhttps://zbmath.org/1496.530032022-11-17T18:59:28.764376Z"Souto Pérez, Silvia"https://zbmath.org/authors/?q=ai:souto-perez.silvia(no abstract)Surfaces in homogeneous manifolds generated by Schwarz reflectionhttps://zbmath.org/1496.530042022-11-17T18:59:28.764376Z"Windemuth, Arthur"https://zbmath.org/authors/?q=ai:windemuth.arthur(no abstract)Contemporary perspectives in differential geometry and its related fields. Proceedings of the 5th international colloquium on differential geometry and its related fields, Veliko Tarnovo, Bulgaria, September 6--10, 2016https://zbmath.org/1496.530052022-11-17T18:59:28.764376ZPublisher's description: This volume contains original papers and announcements of recent results presented by the main participants of the 5th International Colloquium on Differential Geometry and its Related Fields (ICDG2016). These articles are devoted to some new developments on geometric structures on manifolds. Besides covering a broad overview on geometric structures, this volume provides significant information for researchers not only in the field of differential geometry but also in mathematical physics. Since each article is accompanied with detailed explanations, it serves as a good guide for young scientists working in this area.
The articles of this volume will be reviewed individually. For the preceding colloquium see [Zbl 1329.53005].
Indexed articles:
\textit{Arvanitoyeorgos, Andreas; Sakane, Yusuke; Statha, Marina}, Homogeneous Einstein metrics on complex Stiefel manifolds and special unitary groups, 1-20 [Zbl 1387.53057]
\textit{Okuda, Takayuki}, Geodesics of Riemannian symmetric spaces included in reflective submanifolds, 21-32 [Zbl 1387.53068]
\textit{Kasuya, Hisashi}, A differential geometric viewpoint of mixed Hodge structures, 33-51 [Zbl 1387.58006]
\textit{Bejan, Cornelia-Livia; Gül, İlhan}, \(F\)-geodesics on the cotangent bundle of a Weyl manifold, 53-66 [Zbl 1387.53043]
\textit{Ikawa, Osamu}, The geometry of orbits of Hermann type actions, 67-78 [Zbl 1387.53067]
\textit{Hashimoto, Hideya; Ohashi, Misa}, Fundamental relationship between Cartan imbeddings of type a and Hopf fibrations, 79-94 [Zbl 1387.53065]
\textit{Adachi, Toshiaki}, A study on trajectory-horns for Kähler magnetic fields, 95-112 [Zbl 1387.53094]
\textit{Nakova, Galia}, Null curves on the unit tangent bundle of a two-dimensional Kähler-Norden manifold, 113-129 [Zbl 1387.53092]
\textit{Fukada, Yoshitaka}, Horizontal lifts of curves through a Hopf fibration and some examples of Hopf tori, 131-149 [Zbl 1387.53071]
\textit{Nakata, Fuminori}, Homotopy groups of \(G_2/Sp(1)\) and \(G_2/U(2)\), 151-159 [Zbl 1394.57032]
\textit{Matsuzoe, Hiroshi}, Sequential structure of statistical manifolds and its divergence geometry, 161-178 [Zbl 06833820]An approach for designing a surface pencil through a given geodesic curvehttps://zbmath.org/1496.530062022-11-17T18:59:28.764376Z"Atalay, Gülnur Şaffak"https://zbmath.org/authors/?q=ai:atalay.gulnur-saffak"Güler, Fatma"https://zbmath.org/authors/?q=ai:guler.fatma"Bayram, Ergin"https://zbmath.org/authors/?q=ai:bayram.ergin"Kasap, Emin"https://zbmath.org/authors/?q=ai:kasap.eminSummary: In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. In addition, developablity along the common geodesic of the members of surface family are discussed. Finally, we illustrate this method by presenting some examples.Generalised Bianchi permutability for isothermic surfaceshttps://zbmath.org/1496.530072022-11-17T18:59:28.764376Z"Cho, Joseph"https://zbmath.org/authors/?q=ai:cho.joseph"Leschke, Katrin"https://zbmath.org/authors/?q=ai:leschke.katrin"Ogata, Yuta"https://zbmath.org/authors/?q=ai:ogata.yutaAuthors' abstract: Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic surfaces obey Bianchi permutability: for two distinct spectral parameters, the corresponding Darboux transforms have a common Darboux transform which can be computed algebraically. In this paper, we discuss two-step Darboux transforms with the same spectral parameter, and show that these are obtained by a Sym-type construction: All two-step Darboux transforms of an isothermic surface are given, without further integration, by parallel sections of the associated family of the isothermic surface, either algebraically or by differentiation against the spectral parameter.
Reviewer: Ivan C. Sterling (St. Mary's City)Conformal Kaehler Euclidean submanifoldshttps://zbmath.org/1496.530082022-11-17T18:59:28.764376Z"de Carvalho, A."https://zbmath.org/authors/?q=ai:salles-de-carvalho.andre|de-carvalho.a-a|nolasco-de-carvalho.alexandre|de-carvalho.alcides|de-carvalho.andre-f|ponce-de-leon-ferreira-de-carvalho.andre-carlos|de-carvalho.alexandre-luis-trovon"Chion, S."https://zbmath.org/authors/?q=ai:chion.sergio-j"Dajczer, M."https://zbmath.org/authors/?q=ai:dajczer.marcosSummary: Let \(f : M^{2n} \to \mathbb{R}^{2n+\ell}, n \geq 5\), be a conformal immersion into Euclidean space with codimension \(\ell\) where \(M^{2n}\) is a Kaehler manifold of complex dimension \(n\) free of points where all sectional curvatures vanish. For codimension \(\ell = 1\) or \(\ell = 2\) we show that at least locally such a submanifold can always be obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold \(M^{2n}\) into either \(\mathbb{R}^{2n+1}\) or \(\mathbb{R}^{2n+2}\), the latter being a class of submanifolds already extensively studied.Axial curvature cycles of surfaces immersed in \(\mathbb{R}^4\)https://zbmath.org/1496.530092022-11-17T18:59:28.764376Z"Garcia, R."https://zbmath.org/authors/?q=ai:garcia.ronaldo-a"Sotomayor, J."https://zbmath.org/authors/?q=ai:sotomayor.jorge"Spindola, F."https://zbmath.org/authors/?q=ai:spindola.fSummary: In this paper are established integral expressions, in terms of geometric invariants along a closed curve -- a cycle -- of axial curvature, which characterize \textit{hyperbolicity} i.e. \textit{non unitiy} of the derivative for the Poincaré first return map -- holonomy -- at the axial curvatue cycle on a regular surface \(M\) immersed in \(\mathbb{R}^4 \). A proof of the genericity of hyperbolicity is given here. An integral expression for the second derivative in terms of higher order geometric invariants along a non-hyperbolic axial curvature cycle is also established in this paper. This work improves results obtained by the first and second authors.Parabolic hypersurfaces with constant mean curvature in Euclidean spacehttps://zbmath.org/1496.530102022-11-17T18:59:28.764376Z"Hernández, Mario"https://zbmath.org/authors/?q=ai:hernandez.mario"Meléndez, Josué"https://zbmath.org/authors/?q=ai:melendez.josueIn the present paper the authors consider parabolic hypersurfaces with constant mean curvature \(H\) in Euclidean space \(\mathbb R^{n+m}\) \((m,n\geq 2)\). They give classification results in the case when these hypersurfaces are invariant under a group \(\mathrm{O}(m)\times \mathrm{O}(n)\) and its Gauss-Kronecker curvature \(K\) does not change sign. Case \(H\ne 0\) leads to generalizations of circular cylinders, namely \(\mathbb R^m \times \mathbb S^{n-1}(\rho_1)\) and \(\mathbb R^n \times \mathbb S^{m-1}(\rho_2)\) and hyperspheres \(\mathbb S^{m+n-1}(\rho_3)\), where the positive reals \(\rho_1, \rho_2, \rho_3\) are determined by \(H, m\) and \(n\). In case of \(H=0\) the (non-extendable) solutions are cones \(\mathcal{C}_{m,n}\) generated by the profile curve \(y =\sqrt{\frac{n-1}{m-1}} x\) under the action of \(\mathrm{O}(m)\times \mathrm{O}(n)\).
Reviewer: Friedrich Manhart (Wien)Generalized translation hypersurfaces in conformally flat spaceshttps://zbmath.org/1496.530112022-11-17T18:59:28.764376Z"Sousa, P. A."https://zbmath.org/authors/?q=ai:sousa.paulo-alexandre-araujo"Lima, B. P."https://zbmath.org/authors/?q=ai:lima.barnabe-pessoa"Vieira, B. V. M."https://zbmath.org/authors/?q=ai:vieira.b-v-mThe graph of a real function \(f\) defined in some open set of the Euclidean space of dimension \((p+q)\) is said to be a generalized translation graph (GTG) if \(f\) may be expressed as the sum of two independent functions \(\phi\) and \(\psi\) defined in open sets of the Euclidean spaces of dimension \(p\) and \(q\), respectively. In this paper, the authors study the geometry of GTG immersed in Euclidean space equipped with a metric conformal to the Euclidean metric and obtain results that characterize such hypersurfaces. Applying the characterization results, and using ODE solving techniques, they build examples of GTG satisfying geometric properties not valid in relation to the Euclidean metric.
Reviewer: Atsushi Fujioka (Osaka)Entire constant mean curvature graphs in \(\mathbb{H}^2\times\mathbb{R}\)https://zbmath.org/1496.530122022-11-17T18:59:28.764376Z"Folha, Abigail"https://zbmath.org/authors/?q=ai:folha.abigail"Rosenberg, Harold"https://zbmath.org/authors/?q=ai:rosenberg.haroldGraphs having constant mean curvature \(H\) are called \(H\)-graphs. Previously, the only known examples of entire \(H\)-graphs with \(0<H<\frac{1}{2}\) were conformally hyperbolic invariant surfaces. When \(H=0\), the examples are minimal graphs constructed by P. Collin and the second author.
In this paper, for each \(0\leq H<\frac{1}{2}\), the authors construct entire \(H\)-graphs in \(\mathbb H^2\times\mathbb R\) that are parabolic and not invariant by one-parameter groups of isometries of \(\mathbb H^2\times\mathbb R\). Moreover, their asymptotic boundaries are \((\partial_{\infty}\mathbb H^2)\times\mathbb R\).
Reviewer: Atsushi Fujioka (Osaka)Pure rolling motion of hyperquadrics in pseudo-Euclidean spaceshttps://zbmath.org/1496.530132022-11-17T18:59:28.764376Z"Marques, André"https://zbmath.org/authors/?q=ai:marques.andre-l"Leite, Fátima Silva"https://zbmath.org/authors/?q=ai:silva-leite.fatimaSummary: This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of \textit{R. W. Sharpe} [Differential geometry. New York, NY: Springer (1997)] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.3-dimensional Levi-Civita metrics with projective vector fieldshttps://zbmath.org/1496.530142022-11-17T18:59:28.764376Z"Manno, Gianni"https://zbmath.org/authors/?q=ai:manno.gianni"Vollmer, Andreas"https://zbmath.org/authors/?q=ai:vollmer.andreasProjective vector fields are the infinitesimal transformations whose local flow preserves geodesics up to reparametrization.
In 1882, Sophus Lie posed the following problem for 2-dimensional surfaces: Find the metrics that describe surfaces whose geodesic curves admit an infinitesimal transformation. This problem has been solved recently in [\textit{R. L. Bryant} et al., Math. Ann. 340, No. 2, 437--463 (2008; Zbl 1144.53025); \textit{V. S. Matveev}, Math. Ann. 352, No. 4, 865--909 (2012; Zbl 1237.53013); the authors, J. Math. Pures Appl. (9) 135, 26--82 (2020; Zbl 1434.53012)]. The present paper deals with the analogous problem for manifolds of dimension 3. In particular, the authors solve this problem for 3-dimensional metrics and, more generally, for Levi-Civita metrics of arbitrary signature.
Reviewer: Miroslaw Doupovec (Brno)A discrete version of Liouville's theorem on conformal mapshttps://zbmath.org/1496.530152022-11-17T18:59:28.764376Z"Pinkall, Ulrich"https://zbmath.org/authors/?q=ai:pinkall.ulrich"Springborn, Boris"https://zbmath.org/authors/?q=ai:springborn.boris-andreIn the paper under review, a discrete version of Liouville's theorem for simplicial complexes is given and proved using Cauchy's rigidity theorem for convex polyhedra and its higher-dimensional generalization.
Reviewer: Andreea Olteanu (Bucureşti)Legendrian dualities and evolute-involute curve pairs of spacelike fronts in null spherehttps://zbmath.org/1496.530162022-11-17T18:59:28.764376Z"Song, Xue"https://zbmath.org/authors/?q=ai:song.xue"Li, Enze"https://zbmath.org/authors/?q=ai:li.enze"Pei, Donghe"https://zbmath.org/authors/?q=ai:pei.dongheSummary: The curves that may have singularities on the null sphere have not been discussed before. Since they are degenerate, we can not investigate them directly. So we would like to solve these problems by using the duality theory and the results of our previous studies in Euclidean space. In addition, we discuss the evolutes and involutes of spacelike fronts and investigate their geometric properties in null sphere. Then we also give the classifications of the singularities of evolutes and involutes. Moreover, we give the dual relationships between nullcone surfaces and spacelike fronts.A construction of converging Goldberg-Coxeter subdivisions of a discrete surfacehttps://zbmath.org/1496.530172022-11-17T18:59:28.764376Z"Tao, Chen"https://zbmath.org/authors/?q=ai:tao.chenSummary: It is a fundamental problem to find a good approximating mesh of a given smooth surface and to compare geometries of these two. In the present paper, however, we address this problem from the opposite direction, namely, how we can discover an underlying smooth surface, if it exists, for a given discrete surface (3-valent graph) in \(\mathbb{E}^3\). We construct a method to subdivide a discrete surface in the sense of \textit{M. Kotani} et al. [Comput. Aided Geom. Des. 58, 24--54 (2017; Zbl 1381.65013)] as the solution of the Dirichlet problem for a Goldberg-Coxeter subdivision and prove the sequence of subdivisions forms a Cauchy sequence in the Hausdorif topology. As an application, we give a reason for a carbon network, the Mackay crystal [\textit{A. L. Mackay} and \textit{H. Terrones}, ``Diamond from graphite'', Nature 352, 762 (1991)], to be called a discrete Schwarz P surface (triply periodic minimal surface) known in materials science.Almost geodesic mappings and projections of the spherehttps://zbmath.org/1496.530182022-11-17T18:59:28.764376Z"Mikeš, J."https://zbmath.org/authors/?q=ai:mikes.josef"Guseva, N. I."https://zbmath.org/authors/?q=ai:guseva.nadezhda-ivanovna"Peška, P."https://zbmath.org/authors/?q=ai:peska.patrik"Rýparová, L."https://zbmath.org/authors/?q=ai:ryparova.lenkaIn the paper under review it is shown that the parallel and central projections of \(n\)-planes onto \(n\)-spheres, as well as the central projections of \(n\)-spheres from their centers onto \(n\)-spheres, are almost geodesic mappings. Examples of almost geodesics mappings of compact spaces are also constructed.
Reviewer: Miroslaw Doupovec (Brno)Numerical accuracy of ladder schemes for parallel transport on manifoldshttps://zbmath.org/1496.530192022-11-17T18:59:28.764376Z"Guigui, Nicolas"https://zbmath.org/authors/?q=ai:guigui.nicolas"Pennec, Xavier"https://zbmath.org/authors/?q=ai:pennec.xavierSummary: Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. \textit{Ladder} methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild's ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild's ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.Curvatures of spherically symmetric metrics on vector bundleshttps://zbmath.org/1496.530202022-11-17T18:59:28.764376Z"Abbassi, Mohamed Tahar Kadaoui"https://zbmath.org/authors/?q=ai:abbassi.mohamed-tahar-kadaoui"Lakrini, Ibrahim"https://zbmath.org/authors/?q=ai:lakrini.ibrahimSummary: This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion.The isometric embedding problem for length metric spaceshttps://zbmath.org/1496.530212022-11-17T18:59:28.764376Z"Minemyer, Barry"https://zbmath.org/authors/?q=ai:minemyer.barryIn this paper the author proves a generalization of John Nash's theorem on the isometric embedding of a Riemannian manifold into a Euclidean space. The author proves the following statement:
\textbf{Main Theorem}. Let \(\mathcal{X}\) be a proper \(n\)-dimensional length metric space, and let \(\mathcal{D} \subseteq \mathcal{X}\) be any countable dense subset. Then there exists a collection of geodesics \(\Gamma\) associated to \(\mathcal{D}\) and an embedding \(f :\mathcal{X} \rightarrow \mathbb{R}^{3n+6,1}\) such that \(E(\gamma) = E(f \circ \gamma)\) for all \(\gamma \in \Gamma\). Moreover, if desired, the map \(f\) can be constructed so that its projection onto \(\mathbb{R}^{3n+6,1}\) is not locally Lipschitz. The author explains the proof of the theorem in full detail.
In the introduction the author explains the path to prove the theorem dividing it into 4 majors steps. Step 1 consists of writing the set \(\mathcal{D}\) as the increasing union of finite sets (Section 4). Step 2 (Section 5) consists of the proof of a lemma (Lemma 3) that according to the author \textit{may be of its own independent interest}. In the paper Lemma 3 is the case when \(i=1\) for the proof of the main theorem which is recursive.
\textbf{Lemma 3} Let \((\mathcal{X}, d)\) be an \(n\)-dimensional proper length metric space, let \(\mathcal{D} \subseteq \mathcal{X}\) be any finite subset of \(\mathcal{X}\), and let \(\delta > 0\). Then there exists a map \(f :\mathcal{X} \rightarrow \mathbb{E}^{2n+5}\) which satisfies:
\begin{itemize}
\item[1.] The map \(f\) is \(\delta\)-close to being an embedding. That is, \(f\) satisfies \(f(x) = f(x^{\prime}) \Rightarrow d(x, x^{\prime}) < \delta\)
\item[2.] The map \(f\) is an isometry when restricted to \(\mathcal{D}\). That is, \(d_{f(\mathcal{X})}(f(x), f(x^{\prime})) = d_{\mathcal{X}} (x, x^{\prime})\) for all \(x\), \(x^{\prime}\) \(\in \mathcal{D}\).
\end{itemize}
In Step 3 (Section 6) the author constructs open covers \(\Omega _i\) corresponding to the pair \((\mathcal{D}_i, \Gamma_i)\) in such a way that the mesh of \(\Omega _{i+1}\) is strictly less than one third of the Lebesgue number for \(\Omega _i\) for each \(i\). Step 4 is the conclusion of the proof (Section 7). The article includes three more sections besides the introduction. In Section 2 the energy functional on general metric spaces is studied and how it behaves under perturbations of maps. In Section 3 the theory of indefinite metric polyhedra is discussed and in Section 8 the author presents a ``few of the more `well-known' preliminaries'' and proves that: if \(\mathcal{X}\) is a Finsler manifold which is not Riemannian, and there exists a (path) isometric embedding \(f :\mathcal{X} \rightarrow\mathbb{R}^{p,q}\), \(\pi^+\) and \(\pi^-\) denote the natural projections from \(\mathbb{R}^{p,q}\) onto \(\mathbb{R}^{p,0}\) and \(\mathbb{R}^{0,q}\), respectively, then \(p > 0, q >0\), and both maps \(\pi^+\circ f\) and \(\pi^-\circ f\) are not locally Lipschitz. This proposition is important to understand the main theorem. I would like to call the attention to all the information that the author provides in the introduction, including a long list of remarks related to the main theorem.
Reviewer: Ana Pereira do Vale (Braga)The fundamental theorem of Legendrian submanifolds in the Heisenberg grouphttps://zbmath.org/1496.530222022-11-17T18:59:28.764376Z"Chiu, Hung-Lin"https://zbmath.org/authors/?q=ai:chiu.hung-lin"Lai, Sin-Hua"https://zbmath.org/authors/?q=ai:lai.sin-hua"Li, Jian-Wei"https://zbmath.org/authors/?q=ai:li.jianweiSummary: We study uniqueness and existence questions for Legendrian submanifolds in the Heisenberg group via Cartan's method of moving frames and the theory of Lie groups. Moreover, we generalize this result to any dimensional isotropic submanifolds in the Heisenberg group.\(J\)-trajectories in Vaisman manifoldshttps://zbmath.org/1496.530232022-11-17T18:59:28.764376Z"Inoguchi, Jun-ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichi"Lee, Ji-Eun"https://zbmath.org/authors/?q=ai:lee.jieunLocally conformal Kähler manifolds with parallel Lee fields are called Vaisman manifolds. The authors prove that the curvatures along a smooth curve satisfying the J-trajectory equation in general Vaisman manifolds and J-trajectories are Frenet curves of order at most 4. J-trajectories of constant curvatures in the product spaces of Sasakian manifolds and the real line explicitly are determined.
Reviewer: Huili Liu (Shenyang)Hemi-slant submanifold of \((\operatorname{LCS})_n\)-manifoldhttps://zbmath.org/1496.530242022-11-17T18:59:28.764376Z"Karmakar, Payel"https://zbmath.org/authors/?q=ai:karmakar.payel"Bhattacharyya, Arindam"https://zbmath.org/authors/?q=ai:bhattacharyya.arindamSummary: In this paper we analyse briefly some properties of hemi-slant sub-manifold of \((LCS)_n\)-manifold. Here we discuss about some necessary and sufficient conditions for distributions to be integrable and obtain some results in this direction. We also study the geometry of leaves of hemi-slant submanifold of \((LCS)_n\)-manifold. At last we give an example of a hemi-slant submanifold of an \((LCS)_n\)-manifold.Differential invariants of curves in \(G_2\) flag varietieshttps://zbmath.org/1496.530252022-11-17T18:59:28.764376Z"Kruglikov, Boris"https://zbmath.org/authors/?q=ai:kruglikov.boris-s"Llabrés, Andreu"https://zbmath.org/authors/?q=ai:llabres.andreuAuthors' abstract: We compute the algebra of differential invariants of unparametrized curves in the homogeneous \(G_2\) flag varieties, namely in \(G_2 /P\). This gives a solution to the equivalence problem for such curves. We consider the cases of integral and generic curves and relate the equivalence problems for all three choices of the parabolic subgroup \(P\).
Reviewer: Mohammed El Aïdi (Bogotá)Certain invariant submanifolds of generalized Sasakian-space-formshttps://zbmath.org/1496.530262022-11-17T18:59:28.764376Z"Sarkar, Avijit"https://zbmath.org/authors/?q=ai:sarkar.avijit"Biswas, Nirmal"https://zbmath.org/authors/?q=ai:biswas.nirmalSummary: In the present paper we initiate the study of invariant submanifolds of generalized Sasakian-space-forms with K-contact metric. We consider pseudo-parallel and generalized Ricci pseudo-parallel invariant submanifolds of generalized Sasakian-space-forms. Also we characterize invariant submanifolds whose second fundamental form satisfies some restrictions with respect to concircular curvature tensor of the ambient manifold. Finally, we give some examples of invariant submanifolds of generalized Sasakian-space-forms to verify our results.Lightlike and ideal tetrahedrahttps://zbmath.org/1496.530272022-11-17T18:59:28.764376Z"Meusburger, Catherine"https://zbmath.org/authors/?q=ai:meusburger.catherine"Scarinci, Carlos"https://zbmath.org/authors/?q=ai:scarinci.carlosSummary: We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume formula of ideal hyperbolic tetrahedra.Three-dimensional Riemannian manifolds and Ricci solitonshttps://zbmath.org/1496.530282022-11-17T18:59:28.764376Z"Chaubey, Sudhakar K."https://zbmath.org/authors/?q=ai:chaubey.sudhakar-kumar"De, Uday Chand"https://zbmath.org/authors/?q=ai:de.uday-chandThe aim of the article is to characterize the 3-dimensional Riemannian manifolds endowed with a semi-symmetric metric \(\rho\)-connection if its Riemannian metrics are Ricci and gradient Ricci solitons. It is shown that a 3-dimensional Riemannian manifold equipped with a semi-symmetric metric \(\rho\)-connection admits a Ricci soliton, then the manifold is of constant sectional curvature \(-1\) and the soliton is expanding with \(\lambda=-2\). The authors also construct a non-trivial example of a 3-dimensional Riemannian manifold endowed with a semi-symmetric metric \(\rho\) connection admitting a Ricci solition.
Reviewer: Andreas Arvanitoyeorgos (Patras)Covariant derivatives for Ehresmann connectionshttps://zbmath.org/1496.530292022-11-17T18:59:28.764376Z"Prince, G. E."https://zbmath.org/authors/?q=ai:prince.geoffrey-eamonn|prince.geoff-e"Saunders, D. J."https://zbmath.org/authors/?q=ai:saunders.david-j|saunders.dylan-jSummary: We deal with the construction of covariant derivatives for some quite general Ehresmann connections on fibre bundles. We show how the introduction of a vertical endomorphism allows construction of covariant derivatives separately on both the vertical and horizontal distributions of the connection which can then be glued together on the total space. We give applications across an important class of tangent bundle cases, frame bundles and the Hopf bundle.Minimal rational curves and 1-flat irreducible G-structureshttps://zbmath.org/1496.530302022-11-17T18:59:28.764376Z"Hwang, Jun-Muk"https://zbmath.org/authors/?q=ai:hwang.jun-muk"Li, Qifeng"https://zbmath.org/authors/?q=ai:li.qifengIn the paper the authors study irreducible \(G\)-structures admitting torsion-free affine connections in the setting of algebraic geometry, by considering varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. They prove that such a structure has to be locally symmetric when the dimension of the uniruled projective manifold is at least 5.
When the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety they show that without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. To prove the result they use the method of Cartan connections considering the Cartan connections for certain geometric structures on the spaces of minimal rational curves. In particular, they apply a method developed by the authors in [J. Geom. Anal. 32, No. 6, Paper No. 179, 37 p. (2022; Zbl 07505946)] where they described for a general class of geometric structures the obstructions to the existence of Cartan connections as certain holomorphic tensors.
Reviewer: Anna Fino (Torino)The classifying Lie algebroid of a geometric structure. II: \(G\)-structures with connectionhttps://zbmath.org/1496.530312022-11-17T18:59:28.764376Z"Loja Fernandes, Rui"https://zbmath.org/authors/?q=ai:fernandes.rui-loja"Struchiner, Ivan"https://zbmath.org/authors/?q=ai:struchiner.ivanThis work is the second of two papers dedicated to a systematic study of symmetries, invariants and moduli spaces of geometric structures of finite type, and contains part of the reults obtained by the second author in his Ph.D. thesis [The classifiying Lie algebroid of a geometric structure. University of Campinas (PhD Thesis) (2009)].
The first paper [\textit{R. Loja Fernandes} and \textit{I. Struchiner}, Trans. Am. Math. Soc. 366, No. 5, 2419--2462 (2014; Zbl 1285.53018)] was dedicated to the case of \(\{e\}\)-structures and a special case of Cartan's realization problem. The central object introduced was the classifying Lie algebroid of a fully regular coframe. The classifying Lie algebroid contains all the relevant information for the equivalence problem for \(\{e\}\)-structures.
In the work under review the authors do similar analysis for general \(G\)-structures and the general case of Cartan's realization problem. The appropiate language to deal with such problem is the theory of \(G\)-structure groupoids and \(G\)-structure algebroids developed by the authors in [``The global solutions to Cartan's realization problem'', Preprint, \url{arXiv:1907.13614}].
Given a \(G\)-structure on a manifold \(M\), denoted by \(F_{G}(M)\), equipped with a connection \(\omega \in \Omega ^{1}(F_{G}(M),\mathfrak{g})\), where \( F_{G}(M)\) is the Lie algebra of \(G\), let \(\Omega ^{\bullet }(F_{G}(M), \mathfrak{g})\) be the space of invariants forms consisting of all differential forms which are preserved under local equivalences of \(\left( F_{G}(M),\omega \right) \). The \(G\)-structure with connection \(\left( F_{G}(M),\omega \right) \) is said to be fully regular when the space
\[
\left\{ d_{p}I:I\in \Omega ^{0}(F_{G}(M),\omega )\right\} \subset T_{p}^{\ast }F_{G}(M)\text{, }p\in F_{G}(M)
\]
has constant dimension. To fully regular \(G\)-structure with connection \(\left( F_{G}(M),\omega \right) \), there is naturally associated a vector bundle \(A\rightarrow X\) such that
\[
\Omega ^{\bullet }(F_{G}(M),\mathfrak{g})\simeq \Gamma (\wedge ^{\bullet }A^{\ast }).
\]
It follows that \(A\) has a Lie algebroid structure, called the classifying Lie algebroid of \(\left( F_{G}(M),\omega \right) \).
In this work the authors make precise the link between \(G\)-structure algebroids with connection and the classifying Lie algebroid of a single \(G\)-structure. In the last section, they show how \(G\)-realizations of a \(G\)-structure algebroid with connection are related to \(G\)-integration and how they can be used to codify the solutions of the associated realization method of Cartan.
Reviewer: Eugenia Rosado María (Madrid)Explicit soliton for the Laplacian co-flow on a solvmanifoldhttps://zbmath.org/1496.530322022-11-17T18:59:28.764376Z"Moreno, Andrés J."https://zbmath.org/authors/?q=ai:moreno.andres-j"Sá Earp, Henrique N."https://zbmath.org/authors/?q=ai:sa-earp.henrique-nIn the paper under review, solitons are seen as \(G_2\)-structures which scale monotonically under the flow and move by diffeomorphisms, providing potential models for singularities of the flow and tools for desingularising certain singular \(G_2\)-structures.
The author obtains an example of an explicit soliton, or invariant self-similar solution of the Laplacian co-flow on a particular almost abelian 7-manifold, by applying the Ansatz for the Laplacian co-flow of invariant \(G_2\)-structures on a Lie group, from [\textit{J. Lauret}, Geometric flows and their solitons on homogeneous spaces, Rend. Semin. Mat., Univ. Politec. Torino 74, No. 1, 55--93 (2016; Zbl 1440.53061)].
Reviewer: Simona Druta-Romaniuc (Iaşi)Deformations of nearly \(G_2\) structureshttps://zbmath.org/1496.530332022-11-17T18:59:28.764376Z"Nagy, Paul-Andi"https://zbmath.org/authors/?q=ai:nagy.paul-andi"Semmelmann, Uwe"https://zbmath.org/authors/?q=ai:semmelmann.uweA \(G_2\)-structure \(\varphi\) on a \(7\)-manifold \(M\) is called (strictly) nearly \(G_2\) if it satisfies the condition \(d \varphi = \tau_0 * \varphi\), where \(\tau_0\) is a non-zero real number and \(*\) is the Hodge star operator with respect to the metric \(g_{\varphi}\) induced by \(\varphi\). Nearly \(G_2\)-structures are in one-to-one correspondence with Riemannian metrics in dimension 7 admitting Killing spinors. In particular, \(g_{\varphi}\) has to be an Einstein metric of positive scalar curvature.
The main classes of examples admitting nearly \(G_2\)-structures are homogeneous, including the Aloff-Wallach spaces \(N(k,l)\), and the ones obtained from a canonical variation of a 3-Sasaki metric in dimension 7.
In the paper the authors study the deformation theory of nearly \(G_2\)-structures, obtaining a description of the second-order obstruction to deformation on nearly \(G_2\)-structures on compact 7-manifolds. To prove the result they use that infinitesimal deformations of nearly \(G_2\)-structures correspond to the kernel of \(\Delta^{g_{\varphi}} - \tau_0^2\) acting on the intersection \(\Omega^4_{27} (M) \cap \mathrm{Ker} \, d\). As an application they show that all infinitesimal deformations of the Aloff-Wallach space \(N(1,1)\) are obstructed to second order.
Reviewer: Anna Fino (Torino)On the Godbillon-Vey invariant of transversely parallelizable foliationshttps://zbmath.org/1496.530342022-11-17T18:59:28.764376Z"Rovenski, Vladimir"https://zbmath.org/authors/?q=ai:rovenskii.vladimir-yuzefovich"Walczak, Paweł"https://zbmath.org/authors/?q=ai:walczak.pawel-grzegorzThe work enhances the concept of Godbillon-Vey classes of foliations, to a non-integrable and a variational framework.
Let \(M\) be a \((2q+1)\)-dimensional smooth manifold, equipped with a \((q + 1)\)-dimensional, a priori non-integrable, distribution \(\mathcal{D}\) and a \(q\)-vector field \(\mathbf{T} = T_1 \wedge \ldots \wedge T_q\), where \(\{T_i\}\) are linearly independent vector fields transverse to \(\mathcal{D}\).
Using a \(q\)--form \(\omega\) such that \(\mathcal{D} = \hbox{ker} (\omega)\) \, and \(\omega (\mathbf{T}) = 1\), the authors construct a \((2q+1)\)-form analogous to that defining the generalized Godbillon-Vey class of a foliation of codimension \(q\). They show how this form does depend on \(\omega\) and \(\mathbf{T}\).
Moreover, for a compatible Riemannian metric \(g\) on \(M\), the authors express this \((2q + 1)\)-form in terms of \(\mathbf{T}\), and the extrinsic geometry of \(\mathcal{D}\) and its normal distribution \(\mathcal{D}^\perp\). Suitable Euler-Lagrange equations of the associated functionals on a closed manifold are constructed, for a variable pair \((\omega,\mathbf{T})\) on \((M, g)\) and for a variable metric on \((M, \mathcal{D})\). The work studies harmonic distributions \(\mathcal{D}^\perp\) such that \((\omega, \mathbf{T})\) is critical, characterize critical pairs when \(\mathcal{D}\) is integrable, and find sufficient conditions for critical pairs when variations are among foliations. The authors also calculate the index form and consider examples of critical foliations among twisted products.
In particular, the present work extends the study in [the first author et al., Complex Anal. Oper. Theory 13, No. 6, 2917--2937 (2019Z; bl 1435.53025)].
Reviewer: Jesus Muciño Raymundo (Morelia)Principal curvature lines near a partially umbilic point of codimension onehttps://zbmath.org/1496.530352022-11-17T18:59:28.764376Z"Sotomayor, Jorge"https://zbmath.org/authors/?q=ai:sotomayor.jorge"Lopes, Débora"https://zbmath.org/authors/?q=ai:lopes.debora"Garcia, Ronaldo"https://zbmath.org/authors/?q=ai:garcia.ronaldo-aSummary: In this work we study the mutually orthogonal foliations, in oriented three dimensional manifolds \(\mathbb{M}^3 \), whose leaves are the integral curves of the principal curvature direction fields associated to immersions \(\alpha:\mathbb{M}^3\rightarrow\mathbb{R}^4 \). We describe these foliations around their singularities, which occur at points, called partially umbilic, where at least two principal curvatures coincide. Here we extend the contributions of R. Garcia, 1989, 2001, further elaborated by R. Garcia, D. Lopes and J. Sotomayor, 2015, concerning the study of the generic singularity patterns denominated \(D_1\), \(D_2\), \(D_3\), \(D_{12}\) and \(D_{23}\). To this end here we establish the principal configurations in a neighborhood of a partially umbilic point \(D_{13}^1\) which appear generically in one parameter families of hypersurfaces of \(\mathbb{R}^4 \).Structure of graphs of suspended foliationshttps://zbmath.org/1496.530362022-11-17T18:59:28.764376Z"Zhukova, N. I."https://zbmath.org/authors/?q=ai:zhukova.nina-ivanovna"Chubarov, G. V."https://zbmath.org/authors/?q=ai:chubarov.georgy-vIn this paper, the authors study suspended foliations with an Ehresmann connection and the structure of the graphs of these foliations. In [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 16, 367--397 (1962; Zbl 0122.40702)], the notion of a suspended foliation was introduced by \textit{A. Haefliger}. In 1983, Ehresmann introduced the notion of a holonomy groupoid. The equivalent construction, i.e. the notion of the graph of a foliation, was introduced by \textit{H. E. Winkelnkemper} [Ann. Global Anal. Geom. 1, No. 3, 51--75 (1983; Zbl 0526.53039)] and this concept quickly became one of the main constructions in the theory of foliations. The purpose of the paper is to study the structure of the graphs of suspended foliations. The main results of the paper are three theorems which are stated in Section 1. The first result is Theorem 1.1 in which the authors give a necessary and sufficient condition for a foliation to be a suspended one. As a direct consequence of this theorem, the suspended foliations have the natural integrable Ehresmann connection. The second result is Theorem 1.2 and it states that if a given foliation is suspended then the induced foliation on the graph of the original one is also suspended; moreover, the structure group and global holonomy group of these foliations coincide. The last result of the paper is Theorem 1.4 in which they give a detailed description of the structure of graphs of suspended foliations and their induced foliations.
Reviewer: Le Anh Vu (Ho Chi Minh City)Cartan connections and path structures with large automorphism groupshttps://zbmath.org/1496.530372022-11-17T18:59:28.764376Z"Falbel, E."https://zbmath.org/authors/?q=ai:falbel.elisha"Mion-Mouton, M."https://zbmath.org/authors/?q=ai:mion-mouton.martin"Veloso, J. M."https://zbmath.org/authors/?q=ai:veloso.jose-miguel-martinsIn this paper the authors consider compact 3-dimensional manifolds equipped with a Lagrangian contact structure or \textit{path structure}. This consists of a couple of 1-dimensional distributions \((E_1,E_2)\) such that \(E_1\oplus E_2\) is a contact distribution. If a contact form \(\theta\) is fixed, whose kernel is the contact distribution \(E_1\oplus E_2\), then the triplet \(\mathcal{T}=(E_1,E_2,\theta)\) is referred to as a \textit{strict path structure}. A (local) automorphism of \((M,\mathcal{T})\) is a (local) diffeomorphism \(f\) of \(M\) that preserves \(E_1\), \(E_2\) and \(\theta\). The group of such (local) automorphisms is denoted by \(\mathrm{Aut}^{\mathrm{loc}}(M,\mathcal{T})\). The \textit{path structure} is a geometric structure related to the geometry of second-order ordinary differential equations.
In this work, the authors use a description of the strict path geometric structure as a Cartan geometry. The fact that one has a Cartan's connection which is invariant under an automorphism group with a dense orbit implies that some components of its curvature vanish. This allows the classification of all such spaces.
The flat model for strict path geometries is the Heisenberg space \(\mathrm{Heis}(3)\) with two left-invariant directions and a fixed left-invariant contact form. Its automorphism group is \(\mathrm{Heis}(3)\times P\), where \(P\) is a group isomorphic to \(\mathbb{R}^*\). A (non-flat) constant curvature model is given by a left-invariant structure on \(\widetilde{\mathrm{SL}(2,\mathbb{R})}\) (the universal cover of \(\mathrm{SL}(2,\mathbb{R})\)), whose automorphism group is \(\widetilde{\mathrm{SL}(2,\mathbb{R})}\times \widetilde A\), where \(\widetilde A\) is a group isomorphic to \(\mathbb{R}^*\).
The main result of this paper is the following classification theorem.
\textbf{Theorem.} Let \(\mathcal{T}\) be a strict path structure on a compact three-dimensional manifold. If \(\mathcal{T}\) has a non-compact automorphism group (for the compact open topology) and a dense \(\mathrm{Aut}^{\mathrm{loc}}(M,\mathcal{T})\)-orbit, then
(1) either \((M,{\mathcal{T}})\) is, up to a constant multiplication of its contact form, isomorphic to \(\Gamma \setminus \mathrm{SL}(2,\mathbb{R})\) for some discrete subgroup \(\Gamma\) of \(\mathrm{SL}(2,\mathbb{R})\times A\) acting freely, properly and cocompactly on \(\mathrm{SL}(2,\mathbb{R})\);
(2) or \((M,{\mathcal{T}})\) is, up to finite covering, isomorphic to \(\Gamma \setminus\mathrm{Heis}(3),\) for some cocompact lattice of \(\mathrm{Heis}(3)\).
The above theorem is a generalization of \textit{É. Ghys}' theorem classifying contact-Anosov flows with smooth invariant distributions on compact three-manifolds [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 2, 251--270 (1987; Zbl 0663.58025)].
Reviewer: Laura Geatti (Roma)On a Riemannian manifold with two orthogonal distributionshttps://zbmath.org/1496.530382022-11-17T18:59:28.764376Z"Rovenski, Vladimir"https://zbmath.org/authors/?q=ai:rovenskii.vladimir-yuzefovichIn the paper the author studies a Riemannian manifold with a distribution \(\mathcal D\) which splits as an orthogonal sum of two distributions \(\mathcal D=\mathcal D_1\oplus \mathcal D_2\). If \(P:TM\rightarrow \mathcal D\) is an orthoprojection and \(X\in \mathfrak X(M)\) then the P-divergence of \(X\) is defined. The curvature tensor \(R^P\) and mixed scalar curvature \(\mathcal S^P_{\mathcal D_1, \mathcal D_2}\) are defined. The integral formulas for a Riemannian manifold equipped with orthogonal distributions \(\mathcal D_1, \mathcal D_2\) are proven and some of its applications are given.
For the entire collection see [Zbl 1481.26002].
Reviewer: Włodzimierz Jelonek (Kraków)Ricci solitons on para-Sasakian manifolds satisfying pseudo-symmetry curvature conditionshttps://zbmath.org/1496.530392022-11-17T18:59:28.764376Z"Singh, Abhishek"https://zbmath.org/authors/?q=ai:singh.abhishek-kumar|singh.abhishek|singh.abhishek-kr"Kishor, Shyam"https://zbmath.org/authors/?q=ai:kishor.shyamSummary: The paper deals with the study of Ricci solitons on para-Sasakian manifolds satisfying pseudo-symmetry curvature conditions. First, we investigate Ricci solitons in Ricci-pseudosymmetric para-Sasakian manifolds. Next, we consider Ricci solitons in \(W_3\)-Ricci-pseudo-symmetric para-Sasakian manifolds. Moreover, we investigate Ricci solitons in Ricci generalized pseudo-symmetric para-Sasakian manifold. Finally, we prove that Ricci solitons in para-Sasakian manifolds satisfying the curvature condition \(Q \cdot R = 0\), is expanding and an example is given to verify the theorem.Left invariant special Kähler structureshttps://zbmath.org/1496.530402022-11-17T18:59:28.764376Z"Valencia, Fabricio"https://zbmath.org/authors/?q=ai:valencia.fabricioAuthors' abstract: We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.
Reviewer: Mohammed El Aïdi (Bogotá)On the induced geometry on surfaces in 3D contact sub-Riemannian manifoldshttps://zbmath.org/1496.530412022-11-17T18:59:28.764376Z"Barilari, Davide"https://zbmath.org/authors/?q=ai:barilari.davide"Boscain, Ugo"https://zbmath.org/authors/?q=ai:boscain.ugo"Cannarsa, Daniele"https://zbmath.org/authors/?q=ai:cannarsa.danieleLet \(M\) be a 3-dimensional manifold with a contact sub-Riemannian structure \((D,g)\), and let \(d_{sR}\) be the induced sub-Riemannian distance on \(M\). If \(S \subset M\) is a 2-dimensional submanifold (i.e., an embedded surface), there is induced a distance \(d_S\) on \(S\), where \(d_S(x,y)\) is defined as the infimum of the lengths of all horizontal paths that lie in \(S\) and join \(x\) and \(y\). Note that \(d_S\) is not the restriction of \(d_{sR}\) to \(S\), nor is it the sub-Riemannian distance induced by the restriction of \((D,g)\) to \(S\) (which is not bracket-generating). In this paper, the authors study sufficient conditions for \(d_S\) to be finite on \(S\), which is to say that any two points of \(S\) are joined by a finite-length horizontal path lying in \(S\).
The first part of the paper is concerned with local properties of the characteristic foliation induced by the distribution \(D \cap TS\), in a neighborhood of a characteristic point \(p\) (i.e., a point where \(S\) is tangent to \(D\)). The authors introduce a scalar quantity \(\widehat{K}_p\), defined in terms of the Gaussian curvature of \(S\) with respect to Riemannian approximations to the sub-Riemannian geometry \((D,g)\), but which is shown to be independent of the choice of Riemannian approximation (Theorem 1.1.). The value of \(\widehat{K}_p\) turns out to relate to the eigenvalues of \(DX(p)\) (Proposition 1.2), where \(X\) is the characteristic vector field, and thus is useful in classifying the qualitative behavior of the foliation near \(p\) as a saddle, saddle-node, node, or focus (Corollaries 4.5 and 4.7). This is then used to address the question of whether or not a horizontal path in \(S\) that approaches \(p\) will have finite length (Proposition 1.3).
In the second part of the paper (Theorem 1.5), the authors prove that if a compact connected \(C^2\) surface \(S \subset M\) satisfies the following set of conditions, then the induced distance \(d_S\) is finite on \(S\):
\begin{itemize}
\item The contact sub-Riemannian structure \((M,D,g)\) is coorientable (i.e., the distribution \(D\) is the kernel of a globally defined contact one-form \(\omega\)) and is tight (admits no overtwisted disk);
\item The surface \(S\) is homeomorphic to the sphere \(S^2\);
\item The characteristic points of \(S\) are isolated.
\end{itemize}
Reviewer: Nathaniel Eldredge (Storrs)Optimal horizontal joinability on the Engel grouphttps://zbmath.org/1496.530422022-11-17T18:59:28.764376Z"Greshnov, Alexander"https://zbmath.org/authors/?q=ai:greshnov.alexandre|greshnov.aleksandr-valerevichAuthor's abstract: On the Engel group we solve the problem of finding the minimal number of segments of integral lines of left-invariant basis horizontal vector fields necessary for joining an arbitrary pair of points. We prove the best version of the Rashevskii-Chow theorem on the Engel group.
Reviewer: Andreea Olteanu (Bucureşti)Area of intrinsic graphs and coarea formula in Carnot groupshttps://zbmath.org/1496.530432022-11-17T18:59:28.764376Z"Julia, Antoine"https://zbmath.org/authors/?q=ai:julia.antoine"Nicolussi Golo, Sebastiano"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastiano"Vittone, Davide"https://zbmath.org/authors/?q=ai:vittone.davideThe authors consider submanifolds of sub-Riemannian Carnot groups with intrinsic \(C^1\) regularity \((C^1_H )\). The first main result in the present paper is an area formula for \(C^1_H\) intrinsic graphs; as an application, the authors deduce density properties for Hausdorff measures on rectifiable sets. The second main result is a coarea formula for slicing \(C^1_H\) submanifolds into level sets of a \(C^1_H\) function
Reviewer: Peibiao Zhao (Nanjing)An operator related to the sub-Laplacian on the quaternionic Heisenberg grouphttps://zbmath.org/1496.530442022-11-17T18:59:28.764376Z"Wang, Haimeng"https://zbmath.org/authors/?q=ai:wang.haimeng"Wang, Bei"https://zbmath.org/authors/?q=ai:wang.beiThe authors in the present paper study an operator related to the sub-Laplacian on the nonisotropic quaternionic Heisenberg group and construct the fundamental solution for this operator. For the isotropic case, the authors derive the closed form of this solution. The techniques the authors used can be applied to the standard Heisenberg group. The authors also give the connection between this operator and the Heisenberg sub-Laplacian.
Reviewer: Peibiao Zhao (Nanjing)Homogeneous conformal \(C\)-spaces in dimension fourhttps://zbmath.org/1496.530452022-11-17T18:59:28.764376Z"Calviño-Louzao, E."https://zbmath.org/authors/?q=ai:calvino-louzao.esteban"García-Río, E."https://zbmath.org/authors/?q=ai:garcia-rio.eduardo"Gutiérrez-Rodríguez, I."https://zbmath.org/authors/?q=ai:gutierrez-rodriguez.ixchel"Vázquez-Lorenzo, R."https://zbmath.org/authors/?q=ai:vazquez-lorenzo.ramonAuthors' abstract: We classify four-dimensional homogeneous conformal \(C\)-spaces and show that they are conformally Cotton-flat.
Reviewer: Mohammed El Aïdi (Bogotá)Ricci solitons on Lorentz-Sasakian space formshttps://zbmath.org/1496.530462022-11-17T18:59:28.764376Z"Lone, Mehraj Ahmad"https://zbmath.org/authors/?q=ai:lone.mehraj-ahmad"Harry, Idrees Fayaz"https://zbmath.org/authors/?q=ai:harry.idrees-fayazSummary: In this paper, we study Lorentz-Sasakian space forms admitting conformal Ricci soliton and conformal gradient Ricci soliton. We also study the conditions for the soliton to be steady, shrinking and expanding. We also showed that depending on the nature of the structure function of Lorentz-Sasakian space form, the potential function of conformal gradient Ricci soliton is constant.Conformal vector fields and \(\sigma_k\)-scalar curvatureshttps://zbmath.org/1496.530472022-11-17T18:59:28.764376Z"Xu, Xingwang"https://zbmath.org/authors/?q=ai:xu.xingwang"Ye, Jian"https://zbmath.org/authors/?q=ai:ye.jianSummary: We discuss a new relationship between the conformal vector field and the \(\sigma_k\)-scalar curvature for general \(k\ge1\) on a closed manifold. The case \(k=1\) is well known and has been widely used. Several applications of the identities derived for the general case are given.Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifoldshttps://zbmath.org/1496.530482022-11-17T18:59:28.764376Z"Bray, Hubert L."https://zbmath.org/authors/?q=ai:bray.hubert-l"Kazaras, Demetre P."https://zbmath.org/authors/?q=ai:kazaras.demetre-p"Khuri, Marcus A."https://zbmath.org/authors/?q=ai:khuri.marcus-a"Stern, Daniel L."https://zbmath.org/authors/?q=ai:stern.daniel-lSummary: An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.Reverse comparison theorems with upper integral Ricci curvature conditionhttps://zbmath.org/1496.530492022-11-17T18:59:28.764376Z"Chen, Hang"https://zbmath.org/authors/?q=ai:chen.hang.1|chen.hang"Gao, Chaoqun"https://zbmath.org/authors/?q=ai:gao.chaoqunAuthors' abstract: We prove some reverse Laplacian comparison and relative volume comparison results under the situation where one has an integral bound for the part of the Ricci curvature which lies above a prescribed continuous function of the distance parameter. These extend parts of results of \textit{Q. Ding} [Chin. Ann. Math., Ser. B 15, No. 1, 35--42 (1994; Zbl 0798.53048)] and \textit{T. Kura} [Proc. Japan Acad., Ser. A 78, No. 1, 7--9 (2002; Zbl 1082.53038)] from pointwise Ricci curvature to integral Ricci curvature.
Reviewer: Joseph E. Borzellino (San Luis Obispo)Quasi-positive curvature on Bazaikin spaceshttps://zbmath.org/1496.530502022-11-17T18:59:28.764376Z"DeVito, Jason"https://zbmath.org/authors/?q=ai:devito.jason"Sherman, Evan"https://zbmath.org/authors/?q=ai:sherman.evan\textit{Ya. V. Bazaikin} constructed in [Sib. Math. J. 37, No. 6, 1068--1085 (1996; Zbl 0874.53034); translation from Sib. Mat. Zh. 37, No. 6, 1219--1237 (1996)] an infinite series of 13-dimensional compact Riemannian manifolds (in the form of biquotients of compact Lie groups), on which there exist Riemannian metrics of positive curvature. Five integers \(q_i\) are involved in his construction. These parameters are usually written as a five-dimensional vector \(\bar q\). This article considers some special Riemannian metrics (the constructions of which are used by Bazaikin; they are called natural), considering also the cases when the curvature is only non-negative. A Riemannian manifold is called quasi-positively curved if it has non-negative sectional curvature and it has a point for which the sectional curvature of all two-planes at that point is positive. A Riemannian manifold is called almost positively curved if the set of points with all two-planes positively curved is open and dense.
The main result of the article is the following: suppose \(B_{\bar q}\) is a Bazaikin space with natural metrics. Then \(B_{\bar q}\) is quasi-positively curved if and only if \(q\) is not a permutation of \(\pm(1, 1, 1, -1, -3)\) and \(B_{\bar q}\) is almost positively curved if and only if it is strictly positively curved or \(\bar q\) is a permutation of \(\pm (1, 1, 1, 1, -1)\). Also it is proved that up to diffeomorphism, every Bazaikin space admits a metric of quasi-positive curvature.
Reviewer: V. V. Gorbatsevich (Moskva)Diameter estimate for closed manifolds with positive scalar curvaturehttps://zbmath.org/1496.530512022-11-17T18:59:28.764376Z"Fu, Xuenan"https://zbmath.org/authors/?q=ai:fu.xuenan"Wu, Jia-Yong"https://zbmath.org/authors/?q=ai:wu.jiayongSummary: For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a conformal immersion into a sphere, the dependency on the Yamabe constant is not necessary. The power of scalar curvature integral in these diameter estimates is sharp and it occurs at round spheres with canonical metric.Deformations of \(Q\)-curvature. II.https://zbmath.org/1496.530522022-11-17T18:59:28.764376Z"Lin, Yueh-Ju"https://zbmath.org/authors/?q=ai:lin.yueh-ju"Yuan, Wei"https://zbmath.org/authors/?q=ai:yuan.weiThe \(Q\)-curvature is a fourth-order scalar Riemannian invariant that has been deeply studied in the last period because it can be considered as the analogue of the Gaussian curvature, but also of the scalar curvature. The authors investigated in a previous paper [Calc. Var. Partial Differ. Equ. 55, No. 4, Paper No. 101, 29 p. (2016; Zbl 1355.53026)] local stability and rigidity phenomena of \(Q\)-curvature, obtaining some interesting geometric results about \(Q\)-curvature. In this paper, the authors study the volume comparison with respect showing that the volume comparison theorem holds for metrics close to strictly stable positive Einstein metrics, meaning that \(Q\)-curvature can still control the volume of manifolds under certain conditions. The local rigidity of strictly stable Ricci-flat manifolds with respect to \(Q\)-curvature is found, and this proves the non-existence of metrics with positive \(Q\)-curvature near the reference metric.
Reviewer: Marian Ioan Munteanu (Iaşi)Para-complex Norden structures in cotangent bundle equipped with vertical rescaled Cheeger-Gromoll metrichttps://zbmath.org/1496.530532022-11-17T18:59:28.764376Z"Zagane, Abderrahim"https://zbmath.org/authors/?q=ai:zagane.abderrahimSummary: In the paper, a deformation (in the vertical bundle) of the Cheeger-Gromoll metric on the cotangent bundle \(T^\ast M\) over an \(m\)-dimensional Riemannian manifold \((M,g)\), called the vertical rescaled Cheeger-Gromoll metric, is considered. The para-Nordenian properties of the vertical rescaled Cheeger-Gromoll metric are studied.Einstein solitons with unit geodesic potential vector fieldhttps://zbmath.org/1496.530542022-11-17T18:59:28.764376Z"Blaga, Adara M."https://zbmath.org/authors/?q=ai:blaga.adara-monica"Deshmukh, Sharief"https://zbmath.org/authors/?q=ai:deshmukh.shariefThe authors establish some results on almost Einstein solitons with unit geodesic potential vector field and provide necessary and sufficient conditions for the soliton to be trivial. They prove the following contributory result: Let \((g,\xi,\lambda)\) be an almost Einstein soliton on the compact and connected \(n\)-dimensional smooth manifold \(M (n>2)\) with unit geodesic potential vector field \(\xi\) and nonzero scalar curvature. Then \(\xi\) is an eigenvector of the Ricci operator with constant eigenvalue, i.e., \(Q\xi = \sigma\xi\), for \(\sigma\in \mathbb{R}^*\), satisfying \((n\sigma-r)r \geq 0\), if and only if the soliton is trivial.
Reviewer: Mehdi Rafie-Rad (Babolsar)Quasiconformal flows on non-conformally flat sphereshttps://zbmath.org/1496.530552022-11-17T18:59:28.764376Z"Chang, Sun-Yung Alice"https://zbmath.org/authors/?q=ai:chang.sun-yung-alice"Prywes, Eden"https://zbmath.org/authors/?q=ai:prywes.eden"Yang, Paul"https://zbmath.org/authors/?q=ai:yang.paul-c-pSummary: We study integral curvature conditions for a Riemannian metric \(g\) on \(S^4\) that quantify the best bilipschitz constant between \((S^4, g)\) and the standard metric on \(S^4\). Our results show that the best bilipschitz constant is controlled by the \(L^2\)-norm of the Weyl tensor and the \(L^1\)-norm of the \(Q\)-curvature, under the conditions that those quantities are sufficiently small, \(g\) has a positive Yamabe constant and the \(Q\)-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on \(S^4\).New integral estimates in substatic Riemannian manifolds and the Alexandrov theoremhttps://zbmath.org/1496.530562022-11-17T18:59:28.764376Z"Fogagnolo, Mattia"https://zbmath.org/authors/?q=ai:fogagnolo.mattia"Pinamonti, Andrea"https://zbmath.org/authors/?q=ai:pinamonti.andreaSummary: We derive new integral estimates on substatic manifolds with boundary of horizon type, naturally arising in General Relativity. In particular, we generalize to this setting an identity due to [\textit{R. Magnanini} and \textit{G. Poggesi}, J. Anal. Math. 139, No. 1, 179--205 (2019; Zbl 1472.53013)] leading to the Alexandrov Theorem in \(\mathbb{R}^n\) and improve on a Heintze-Karcher type inequality due to [\textit{J. Li} and \textit{C. Xia}, J. Differ. Geom. 113, No. 3, 493--518 (2019; Zbl 1433.53059)]. Our method relies on the introduction of a new vector field with nonnegative divergence, generalizing to this setting the P-function technique of \textit{H. F. Weinberger} [Arch. Ration. Mech. Anal. 43, 319--320 (1971; Zbl 0222.31008)].Comparison theorem and integral of scalar curvature on three manifoldshttps://zbmath.org/1496.530572022-11-17T18:59:28.764376Z"Zhu, Bo"https://zbmath.org/authors/?q=ai:zhu.boSummary: In this note, we extend a comparison theorem of minimal Green functions in [\textit{O. Munteanu} and \textit{J. Wang}, ``Comparison theorems for three-dimensional manifolds with scalar curvature bound'', Preprint, \url{arxiv:2105.12103}] to harmonic functions on complete non-compact three-dimensional manifolds with compact connected boundary. This yields an upper bound on the integral related to scalar curvature on complete, non-parabolic three-dimensional manifolds.Closed geodesics on reversible Finsler 2-sphereshttps://zbmath.org/1496.530582022-11-17T18:59:28.764376Z"De Philippis, Guido"https://zbmath.org/authors/?q=ai:de-philippis.guido"Marini, Michele"https://zbmath.org/authors/?q=ai:marini.michele"Mazzucchelli, Marco"https://zbmath.org/authors/?q=ai:mazzucchelli.marco"Suhr, Stefan"https://zbmath.org/authors/?q=ai:suhr.stefanLet \((M,F)\) be a closed oriented surface endowed with a reversible Finsler metric \(F\), \(\mathbb S^1\) be the unit circle, and \(\mathrm{Emb}(\mathbb S^1,M)\) be the set of smooth \(M\)-valued embedded loops \(\gamma\) on \(\mathbb S^1\). Given \(J\in \mathrm{End}(TM)\), the positive normal to \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) is defined by \(N_{\gamma_t}(u)=\displaystyle\frac{J_{\dot{\gamma_t}}(u)}{\|\dot{\gamma_t}(u)\|}\cdot\) The authors consider a one-parameter family of curves \(\gamma_t\in \mathrm{Emb}(\mathbb S^1,M)\) solutions of
\[\partial_t\gamma_t(u)=\omega_t(u)N_{\gamma_t}(u),\tag{1}\]
where \(\omega_t(u)\) is an explicit expression given in terms of the partial derivatives of \(F\) with respect to some local coordinates on \(M\), \(N_{\gamma_t}(u)\), and on the norm of \(\dot{\gamma_t}\). Then, they state that if \(\gamma_t\) is the solution of (1) with initial condition \(\gamma_0\), then there is a unique \(\mathrm{Emb}(\mathbb S^1,M)\)-valued continuous map \(\varphi\) on an open neighborhood of \(\{0\}\times\mathrm{Emb}(\mathbb S^1,M)\) such that \(\varphi(t,\gamma_0)=\varphi_t(\gamma_0)=\gamma_t\). Furthermore, \(\varphi_t(\gamma\circ\theta)=\varphi_t(\gamma)\circ\theta\) for \(\theta\in Diff(\mathbb S^1)\), \(\frac{d}{dt}L(\varphi_t(\gamma))\le 0\) for \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) where \(L\) is the Finsler length functional on \(\mathrm{Emb}(\mathbb S^1,M)\) defined as \(L(\gamma)=\displaystyle\int_0^1F(\gamma(t),\dot{\gamma_t}(u))du\), and if \(l_\gamma=\displaystyle\lim_{t\to\tau_\gamma}L(\varphi_t(\gamma))>0\), then \(\tau_\gamma=\infty\) (Theorem 2.1). Also, they state that every reversible Finsler two-sphere \((\mathbb S^2,F)\) has at least three explicit geometrically distinct simple closed geodesics (Theorem 1.3).
Reviewer: Mohammed El Aïdi (Bogotá)Intrinsic flat convergence of points and applications to stability of the positive mass theoremhttps://zbmath.org/1496.530592022-11-17T18:59:28.764376Z"Huang, Lan-Hsuan"https://zbmath.org/authors/?q=ai:huang.lan-hsuan"Lee, Dan A."https://zbmath.org/authors/?q=ai:lee.dan-a"Perales, Raquel"https://zbmath.org/authors/?q=ai:perales.raquelA major result in the paper shows that if spaces whose boundaries converge in the Gromov-Hausdorff sense converge in the intrinsic flat sense, then the intrinsic flat convergence and the Gromov-Hausdorff convergence are ``realized'' in the same embedding space. Then the authors use the results to fill in missing details in the proofs of Theorems 1.4 and Lemma 5.1 and address an acknowledged error in the proof of Theorem 1.3 of [the first author et al., J. Reine Angew. Math. 727, 269--299 (2017; Zbl 1368.53028)] by giving a new proof using results claimed in [\textit{B. Allen} and the third author, ``Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below'', Preprint, \url{arXiv:2006.13030}]. One of the aims of the paper is to rectify their previous results in [the first author et al., J. Reine Angew. Math. 727, 269--299 (2017; Zbl 1368.53028)].
Reviewer: Jialong Deng (Beijing)New characterizations of the Whitney spheres and the contact Whitney sphereshttps://zbmath.org/1496.530602022-11-17T18:59:28.764376Z"Hu, Zejun"https://zbmath.org/authors/?q=ai:hu.zejun"Xing, Cheng"https://zbmath.org/authors/?q=ai:xing.chengThe authors of this paper obtain in the first part, an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms of holomorphic curvature \(4c\) with almost complex structure \(J\) that involves the Ricci curvature in the direction \(J\vec{H}\) (\(\vec{H}\) is the mean curvature vector field ) and the norm of the covariant differentiation of the second fundamental form \(h\) and for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure \((\varphi , \xi , \eta , g)\) and constant \(\varphi\) sectional curvature that involves the Ricci curvature in the direction \(\varphi \vec{H}\) and the norm of the modified covariant differentiation of the second fundamental form in the second part. The equality holds if and only if either the Lagrangian submanifolds (Legendrian submanifolds) have parallel second fundamental form or it is one of the Whitney spheres in the complex space forms (respectively C-parallel second fundamental form or is one of the contact Whitney spheres in the Sasakian space forms). They obtain new and global characterizations for the Whitney spheres (for the contact Whitney spheres) in complex space forms (respectively, in Sasakian space forms). The authors prove an similar result for the Whitney spheres in the complex space forms as in [\textit{I. Castro} et al., Pac. J. Math. 199, No. 2, 269--302 (2001; Zbl 1057.53063); \textit{A. Ros} and \textit{F. Urbano}, J. Math. Soc. Japan 50, No. 1, 203--226 (1998; Zbl 0906.53037)], that the contact Whitney spheres are locally conformally flat manifolds and with non-constant sectional curvatures.
Reviewer: Mohamed Belkhelfa (Mascara)Gaussian free fields and Riemannian rigidityhttps://zbmath.org/1496.530612022-11-17T18:59:28.764376Z"Nguyen Viet Dang"https://zbmath.org/authors/?q=ai:nguyen-viet-dang.Summary: On a compact Riemannian manifold \((M,g)\) of dimension \(d\leqslant 4\), we present a rigorous construction of the renormalized partition function \(Z_g(\lambda)\) of a massive Gaussian free field where we explicitly determine the local counterterms using microlocal methods. Then we show that \(Z_g(\lambda)\) determines the Laplace spectrum of \((M,g)\) and hence imposes some strong geometric constraints on the Riemannian structure of \((M,g)\). From this observation, using classical results in Riemannian geometry, we illustrate how the partition function allows us to probe the Riemannian structure of the underlying manifold \((M,g)\).Branched holomorphic Cartan geometry on Sasakian manifoldshttps://zbmath.org/1496.530622022-11-17T18:59:28.764376Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Dumitrescu, Sorin"https://zbmath.org/authors/?q=ai:dumitrescu.sorin"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgSummary: We extend the notion of (branched) holomorphic Cartan geometry on a complex manifold to the context of Sasakian manifolds. Branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds are investigated in particular.The Ricci soliton equation for homogeneous Siklos spacetimeshttps://zbmath.org/1496.530632022-11-17T18:59:28.764376Z"Calvaruso, Giovanni"https://zbmath.org/authors/?q=ai:calvaruso.giovanniSummary: We complete the classification of Ricci solitons within all classes of homogeneous Siklos metrics.On generalized quasi-Einstein manifoldshttps://zbmath.org/1496.530642022-11-17T18:59:28.764376Z"Freitas Filho, Antonio Airton"https://zbmath.org/authors/?q=ai:freitas-filho.antonio-airton"Tenenblat, Keti"https://zbmath.org/authors/?q=ai:tenenblat.ketiSummary: A rigidity result for a class of compact generalized quasi-Einstein manifolds with constant scalar curvature is obtained. Moreover, under some geometric assumptions, the rigidity for the non-compact case is also proved. Considering non constant scalar curvature, we characterize the generalized quasi-Einstein manifolds which is conformal to the Euclidean space and we show that there exist two classes of complete manifolds, which are obtained by considering potential functions and conformal factors either to be radial or invariant under the action of an (n-1) dimensional translation group. Explicit examples are given.Homogeneous Einstein metrics and local maxima of the Hilbert actionhttps://zbmath.org/1496.530652022-11-17T18:59:28.764376Z"Lauret, Jorge"https://zbmath.org/authors/?q=ai:lauret.jorge"Will, Cynthia E."https://zbmath.org/authors/?q=ai:will.cynthia-eSummary: In this short note, three infinite families of neutrally stable homogeneous Einstein metrics are ruled out as candidates for local maxima of the Hilbert action.On some classes of weakly M-projectively symmetric manifoldshttps://zbmath.org/1496.530662022-11-17T18:59:28.764376Z"Pal, Prajjwal"https://zbmath.org/authors/?q=ai:pal.prajjwal"Kar, Debabrata Kumar"https://zbmath.org/authors/?q=ai:kar.debabrata-kumarSummary: The object of the present paper is to study weakly M-projectively symmetric manifolds. At first some geometric properties of \(\text{(WMPS)}_n\) \((n >2)\) have been studied. Next we consider the decomposability of \(\text{(WMPS)}_n\). Finally, we consider \(\text{(WMPS)}_4\) spacetime.Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifoldshttps://zbmath.org/1496.530672022-11-17T18:59:28.764376Z"Peruyero, H. Contreras"https://zbmath.org/authors/?q=ai:peruyero.h-contreras"Suárez-Serrato, P."https://zbmath.org/authors/?q=ai:suarez-serrato.pabloSummary: We show that a combination of collapsing and excessive growth from the fundamental group impedes the existence of Einstein metrics on several families of smooth 4-manifolds. These include infrasolvmanifolds whose fundamental group is not virtually nilpotent, most elliptic surfaces of zero Euler characteristic, geometrizable manifolds with hyperbolic factor geometries in their geometric decomposition, and higher graph 4-manifolds without purely negatively curved pieces.Segre quartic surfaces and minitwistor spaceshttps://zbmath.org/1496.530682022-11-17T18:59:28.764376Z"Honda, Nobuhiro"https://zbmath.org/authors/?q=ai:honda.nobuhiroTwistor spaces are complex manifolds containing certain families of complex curves which are parametrized by manifolds that carry interesting geometric structures. Indeed, twistor spaces were originally invented as a tool to study Riemannian four-manifolds carrying a self-dual Einstein metric. In this case, the twistor space is three-dimensional and the self-dual Einstein four-manifold is recovered as the parameter space of a family of smooth, rational curves with normal bundle \(\mathcal O(1)\oplus \mathcal O(1)\) satisfying a reality condition.
\textit{N.~J.~Hitchin} [Lect. Notes Math. 970, 79--99 (1982; Zbl 0507.53025)] uncovered a similar twistor correspondence between certain projective surfaces containing a smooth, rational curve with self-intersection number two. These surfaces are known as minitwistor spaces and it is known that the deformations of the rational curve, which is called a minitwistor line, are parametrized by a Zariski-open subset \(W_0\) of a smooth, projective three-fold \(W\), carrying a so-called (complex) Einstein-Weyl structure. This theory, in turn, was generalized by the author and \textit{F.~Nakata} [Ann.~Global Anal.~Geom.~39, No.~3, 293--323 (2011; Zbl 1222.53053)] to allow the minitwistor line to have \(g\) nodes and self-intersection \(2+2g\), allowing for more examples. The minitwistor line is a degeneration of smooth curves of genus \(g\), hence \(g\) is called the genus of the minitwistor space.
In this article, the author studies the case \(g=1\), identifying the projective three-fold \(W\) as the (projective) dual of the minitwistor space. This is then used to show that essential minitwistor spaces of genus one are precisely the so-called Segre quartic surfaces. This is a classically known class of surfaces, which arise as complete intersections of two quadrics in \(\mathbb C\mathrm P^4\). The author uses this description to study minitwistor spaces and their dual varieties, making use of the classification of Segre surfaces. Particular attention is paid to the complement of the Weyl-Einstein space \(W_0\) inside \(W\), whose two-dimensional components the author calls divisors at infinity.
Reviewer: Daniel Thung (Hamburg)Classification of Cartan embeddings which are austere submanifoldshttps://zbmath.org/1496.530692022-11-17T18:59:28.764376Z"Kimura, Taro"https://zbmath.org/authors/?q=ai:kimura.taro.1"Mashimo, Katsuya"https://zbmath.org/authors/?q=ai:mashimo.katsuyaAustere submanifolds where introduced by \textit{R. Harvey} and \textit{H. B. Lawson} [Acta Math. 148, 47--157 (1982; Zbl 0584.53021)] in connection with their study of calibrated geometries. In the paper under review the authors classify automorphisms of finite order on compact connected simple Lie groups by which the induced Cartan embedding is an austere embedding. Furthermore, they consider the problem whether an austere Cartan embedding is weakly reflective or not.
Reviewer: Sergei S. Platonov (Petrozavodsk)On decomposition of generalized Riemannian symmetric spaces and complete lifts of Clifford translationshttps://zbmath.org/1496.530702022-11-17T18:59:28.764376Z"Arab, Gholam Hossein"https://zbmath.org/authors/?q=ai:arab.gholam-hossein"Toomanian, Megerdich"https://zbmath.org/authors/?q=ai:toomanian.megerdichSummary: \textit{O. Kowalski} [Generalized symmetric spaces. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0431.53042)], proved that for Riemannian manifolds \(M_1\) and \(M_2\), if their Riemannian direct product \(M={{M}_1}\times{{M}_2}\) is Riemannian symmetric spaces, then \(M_1\) and \(M_2\) are Riemannian symmetric spaces. This paper examines this theorem in generalized symmetric spaces, without assuming that each \(M_i\) is simply connected.Sharp Hardy inequalities via Riemannian submanifoldshttps://zbmath.org/1496.530712022-11-17T18:59:28.764376Z"Chen, Yunxia"https://zbmath.org/authors/?q=ai:chen.yunxia"Leung, Naichung Conan"https://zbmath.org/authors/?q=ai:leung.naichung-conan"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei|zhao.wei.4|zhao.wei.6|zhao.wei.5|zhao.wei.3|zhao.wei.2|zhao.wei.1Summary: This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy inequalities in the cases when the submanifold is compact as well as non-compact. In particular, these inequalities remain valid even if the ambient manifold is compact, in which case we find an optimal space of smooth functions to study Hardy inequalities. Further examples are also provided. Our results complement in several aspects those obtained recently in the Euclidean and Riemannian settings.Moment maps and isoparametric hypersurfaces of OT-FKM typehttps://zbmath.org/1496.530722022-11-17T18:59:28.764376Z"Miyaoka, Reiko"https://zbmath.org/authors/?q=ai:miyaoka.reikoThis short paper complements the results of [the author, Math. Ann. 355, No. 3, 1067-1084 (2013; Zbl 1267.53057)] by giving a new description of the Cartan-Münzner polynomial appearing there in terms of moment maps.
Isoparametric hypersurfaces of the standard Riemannian sphere \(S^{n+1}\) are hypersurfaces with constant principal curvatures. It is known that all such isoparametric hypersurfaces are given by a level set of the so-called Cartan-Münzner polynomial \(F(x)\) in \(\mathbb{R}^{n+2}\) restricted to the sphere. The paper focuses on the case of four distinct principal curvature. The main result is Theorem 5.2: setting \(n = 2l- 2\), it gives a formula for \(F(x)\) in terms of the moment map \(\mu\) -- in the sense of symplectic geometry -- of an action of the Spin group on the higher-dimensional space \(\mathbb{C}^{2l}\). Further, for isoparametric hypersurfaces of OT-FKM type (thus not homogeneous ones), in Theorem 6.1. the author shows that the image of the Gauss map of such a hypersurfaces lies inside the zero level set of a moment map suitably induced by \(\mu\).
Reviewer: Marco Zambon (Leuven)On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz-Minkowski spacetime \(\mathbb{L}^m\)https://zbmath.org/1496.530732022-11-17T18:59:28.764376Z"Palomo, Francisco J."https://zbmath.org/authors/?q=ai:palomo.francisco-j"Romero, Alfonso"https://zbmath.org/authors/?q=ai:romero.alfonsoAccording to the Reilly inequality [\textit{R. C. Reilly}, Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)] for a compact submanifold in the Euclidean space \(\mathbf{E}^m\), it was known that the first non-trivial eigenvalue \(\lambda_1\) of the Laplacian of the induced metric on \(M\) satisfies the following inequality:
\[\lambda_1\geq n\frac{\int_{M}||\mathbf{H}||^{2}dV}{\text{Vol}(M)}.\] The equality holds if and only if \(M\) lies minimally in some hypersphere in \(\mathbf{E}^m\). \newline In this paper, the authors provide an alternate inequality for any compact space-like submanifold \(M\) of the Lorentz-Minkowski spacetime \(\mathbf{L}^m\). Their results show that for each unit time-like vector \(a\in \mathbf{L}^m\), the first nontrivial eigenvalue \(\lambda_1\) satisfies \[\lambda_1\leq n\frac{\int_{M}(||\mathbf{H}||^2+\langle\mathbf{H},a\rangle^2)dV}{\text{Vol}(M)+1/n\int_{M}||a^{T}||^2dV},\] where \(a^T\) is the orthogonal projection of the vector \(a\) on \(T_pM\). Several interesting results are provided along with a clear geometric meaning. By interpreting a compact submanifold \(M\) of a Euclidean space as a compact space-like submanifold of Lorentz-Minkowski spacetime through a space-like hyperplane, the original Reilly inequality is proved also.
Reviewer: Yun Myung Oh (Berrien Springs)Boundary expansions for constant mean curvature surfaces in the hyperbolic spacehttps://zbmath.org/1496.530742022-11-17T18:59:28.764376Z"Han, Qing"https://zbmath.org/authors/?q=ai:han.qing|han.qing.1"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.5|wang.yue.6|wang.yue.1|wang.yue.4|wang.yue.2|wang.yue.3In this paper, the authors study expansions of constant mean curvature hypersurfaces in the hyperbolic space near the boundary and they investigate differences and similarities between the nonzero mean curvature case and the zero mean curvature case. They introduce the differential equation for constant mean curvature hypersurfaces in the hyperbolic space near infinity given by \[ \Delta u -\frac{ u_i u_j}{ 1 + |Du|^2}u_{i j} - \frac{n}{x_n}(u_n- H\sqrt{1 + |Du|^2}) = 0, \text{ in } B^+_1, \]
and \[u = \phi \text{ on } B'_1,\] where \(\phi\) is a given function on \(B'_1\) and \(H\) a constant in \(B^+_1\). The graph of \(u\) for \(\lbrace x_n > 0\rbrace\) is a hypersurface in \(\mathbb{H}^{n+1}\) with its mean curvature given by \(H\) and its asymptotic boundary given by the graph of \(\phi\). The authors discuss the boundary regularity of \(u\) by expanding \(u\) in terms of \(x_n\) and also discuss a general case and allow \(H\) to be a function in \(\bar{B}^+_1\). And finally they discuss the relation between \(u\) and its formal expansions for boundary values of finite regularity and derive sharp estimates of remainders for the asymptotic expansions.
Reviewer: Ameth Ndiaye (Dakar)The Lawson surfaces are determined by their symmetries and topologyhttps://zbmath.org/1496.530752022-11-17T18:59:28.764376Z"Kapouleas, Nikolaos"https://zbmath.org/authors/?q=ai:kapouleas.nikolaos"Wiygul, David"https://zbmath.org/authors/?q=ai:wiygul.davidSummary: We prove that a closed embedded minimal surface in the round three-sphere which satisfies the symmetries of a Lawson surface and has the same genus is congruent to the Lawson surface.Linear Weingarten spacelike submanifolds in semi-Riemannian space formhttps://zbmath.org/1496.530762022-11-17T18:59:28.764376Z"Yu, J. C."https://zbmath.org/authors/?q=ai:yu.jinchao|yu.jin-cheng|yu.jinchang|yu.jichang|yu.jiancheng|yu.jiecheng|yu.jih-chieh|yu.jen-chih|yu.jingcun|yu.j-chulsoo|yu.junche|yu.jinchen|yu.jicheng|yu.jiangchenSummary: We consider the linear Weingarten spacelike submanifold immersed in semi-Riemannian space form \(\mathbb{N}^{n+p}_q (c)\) of constant sectional curvature \(c\) and index \(q\), where \(p \ge q>1\). In this paper, our approach involves an appropriately generalized maximum principle for a suitable Cheng-Yau modified operator so as to obtain that the submanifold is definitely either totally umbilical or isometric to a product space.On conformal Riemannian maps whose total manifold admits a Ricci solitonhttps://zbmath.org/1496.530772022-11-17T18:59:28.764376Z"Gupta, Garima"https://zbmath.org/authors/?q=ai:gupta.garima"Sachdeva, Rashmi"https://zbmath.org/authors/?q=ai:sachdeva.rashmi"Kumar, Rakesh"https://zbmath.org/authors/?q=ai:kumar.rakesh"Rani, Rachna"https://zbmath.org/authors/?q=ai:rani.rachnaSummary: We study conformal Riemannian maps between the Riemannian manifolds. We derive a Bochner type identity and conditions for such maps to be harmonic. Later, we study conformal Riemannian maps whose total manifold admits a Ricci soliton and present a non-trivial example of such conformal Riemannian maps. We also obtain conditions for fiber and range space of such maps to be Ricci soliton and Einstein. We derive conditions for conformal Riemannian maps whose total manifold admits a Ricci soliton to be harmonic and biharmonic.A note on the nonexistence of a complex threefold as a conjugate orbit of \(G_2\)https://zbmath.org/1496.530782022-11-17T18:59:28.764376Z"Guan, Daniel"https://zbmath.org/authors/?q=ai:guan.daniel-zhuang-dan"Wang, Zhonghua"https://zbmath.org/authors/?q=ai:wang.zhonghuaSummary: The existence or nonexistence of a complex structure on \(\mathrm{S}^6\) was a long standing unsolved problem. There is a well-known orbit \(O(\varLambda)\) in \(G_2\) which is diffeomorphic to \(\mathrm{S}^6\) and used by Gábor Etesi in an effort to find a complex structure. Etesi suggested to give a complex structure in \(\mathrm{S}^6\) through this orbit. In Daniel Guan's earlier paper, he proved that the orbit can not be a complex submanifold. Since there was not a clear description of the map from \(O(\varLambda)\) to \(\mathrm{S}^6\) in that paper, we give another clearer, simpler and explicit proof of that result in this paper.Projectively equivalent Finsler metrics on surfaces of negative Euler characteristichttps://zbmath.org/1496.530792022-11-17T18:59:28.764376Z"Lang, Julius"https://zbmath.org/authors/?q=ai:lang.juliusThe paper deals with projectively equivalent Finsler metrics on surfaces of negative Euler characteristic. The main result lies in Theorem 1 where the author proves that on a connected closed surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form. The author proves some small results also in order to prove the main result.
Reviewer: Gauree Shanker (Bathinda)Curvature conditions for spatial isotropyhttps://zbmath.org/1496.530802022-11-17T18:59:28.764376Z"Tzanavaris, Kostas"https://zbmath.org/authors/?q=ai:tzanavaris.kostas"Amaro Seoane, Pau"https://zbmath.org/authors/?q=ai:amaro-seoane.pauSummary: In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalized) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalizes to a class of spaces of non-constant curvature the theorem that spaces of the same constant curvature, dimension and metric tensor sign must be locally isometric. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.Every symplectic manifold is a (linear) coadjoint orbithttps://zbmath.org/1496.530812022-11-17T18:59:28.764376Z"Donato, Paul"https://zbmath.org/authors/?q=ai:donato.paul"Iglesias-Zemmour, Patrick"https://zbmath.org/authors/?q=ai:iglesias-zemmour.patrickRecall that the Kirillov-Kostant-Souriau theorem assures that a symplectic manifold that is homogeneous under the action of a Lie group is isomorphic, up to a covering, to a possibly affine coadjoint orbit.
The authors generalize this result. They prove that every symplectic manifold is a coadjoint orbit of the diffeological group of automorphisms of its integration bundle. The integration bundle is a principal fiber bundle over the manifold, with group the torus of periods of the symplectic form, quotient of the real line by the group of periods, i.e., the integrals of the two-form on every two-cycle.
Reviewer: Daniele Angella (Firenze)On topological properties of positive complexity one spaceshttps://zbmath.org/1496.530822022-11-17T18:59:28.764376Z"Sabatini, S."https://zbmath.org/authors/?q=ai:sabatini.silvia"Sepe, D."https://zbmath.org/authors/?q=ai:sepe.danieleSummary: Motivated by work of Fine and Panov, and of Lindsay and Panov, we prove that every closed symplectic complexity one space that is positive (e.g., positive monotone) enjoys topological properties that Fano varieties with a complexity one holomorphic torus action possess. In particular, such spaces are simply connected, have Todd genus equal to one and vanishing odd Betti numbers.On cosymplectic dynamics. I.https://zbmath.org/1496.530832022-11-17T18:59:28.764376Z"Tchuiaga, Stephane"https://zbmath.org/authors/?q=ai:tchuiaga.stephane"Houenou, Franck"https://zbmath.org/authors/?q=ai:houenou.franck-djideme"Bikorimana, Pierre"https://zbmath.org/authors/?q=ai:bikorimana.pierreThe paper under review is an introduction to cosymplectic topology. By adapting methods from symplectic topology, the authors characterize and study several subgroups of diffeomophisms of a cosymplectic manifold.
Recall that a cosymplectic structure on a smooth manifold is given by a closed \(2\)-form \(\omega\) and a closed \(1\)-form \(\eta\) such that \(\eta \wedge \omega\) is a nowhere vanishing top-form.
In particular, among many other results, the authors show that the Reeb vector field determines the almost cosymplectic nature of a uniform limit of a sequence of almost cosymplectic diffeomorphisms. They also define and study the cosymplectic setting of Hofer and Hofer-like geometries with respect to the group of all cosymplectic diffeomorphisms isotopic to the identity map.
Reviewer: Daniele Angella (Firenze)Links in overtwisted contact manifoldshttps://zbmath.org/1496.530842022-11-17T18:59:28.764376Z"Chatterjee, Rima"https://zbmath.org/authors/?q=ai:chatterjee.rimaSummary: We prove that Legendrian and transverse links in overtwisted contact structures having overtwisted complements can be classified coarsely by their classical invariants. We further prove that any coarse equivalence class of loose links has support genus zero and construct examples to show that the converse does not hold.A note on the infinite number of exact Lagrangian fillings for spherical spunshttps://zbmath.org/1496.530852022-11-17T18:59:28.764376Z"Golovko, Roman"https://zbmath.org/authors/?q=ai:golovko.romanSummary: We discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.Lagrangian pairs of pantshttps://zbmath.org/1496.530862022-11-17T18:59:28.764376Z"Matessi, Diego"https://zbmath.org/authors/?q=ai:matessi.diegoSummary: We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of an exact one form on the real blowup of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of \((\mathbb{C}^*)^n\) that lift tropical subvarieties in \(\mathbb{R}^n\). As an example we explain how to lift tropical curves in \(\mathbb{R}^2\) to Lagrangian submanifolds of \((\mathbb{C}^*)^2\). We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.A monotone Lagrangian casebookhttps://zbmath.org/1496.530872022-11-17T18:59:28.764376Z"Smith, Jack"https://zbmath.org/authors/?q=ai:smith.jackIn this paper the author provides various explicit calculations in Lagrangian Floer theory. This is an interesting and useful collection of results in symplectic reduction, grading periodicity and the closed-open map. As an illustration, here we mention two results.
For each sequence \(k_1,\dots, k_r\) of positive integers, a Lagrangian embedding of the flag variety \(F(d_1, \dots, d_r)\), \(d_j=k_1+\dots+k_j\), in the monotone product of Grassmannians \(\operatorname{Gr}(k_1,n)\times\dots\times \operatorname{Gr}(k_r,n)\), \(n=d_r\), endowed with a specific choice of relative spin structure and the trivial flat line bundle have been constructed and the corresponding Floer cohomology is calculated.
Let \(X\) be a symplectic manifold with a Hamiltonian action of a compact connected Lie group \(K\) such that \(K\) acts freely on the zero level set of the moment map \(\mu^{-1}(0)\). It is proved that if a \(K\)-invariant monotone Lagrangian submanifold \(L\) is a subset of \(\mu^{-1}(0)\), then the induced Lagrangian space \(L/K\) is monotone in the symplectic reduced space \(\mu^{-1}(0)/K\). In particular, \(\mu^{-1}(0)/K\) is (spherically) monotone.
Reviewer: Božidar Jovanović (Beograd)Integrability of quotients in Poisson and Dirac geometryhttps://zbmath.org/1496.530882022-11-17T18:59:28.764376Z"Álvarez, Daniel"https://zbmath.org/authors/?q=ai:alvarez.danielThis paper studies the integrability of Poisson structures (and, more in general, Lie algebroids) that arise from quotient constructions. In this sense it generalises and unifies some previous integrability results that appeared in [\textit{H. Bursztyn} et al., ``Dirac geometry and integration of Poisson homogeneous spaces'', Preprint, \url{arxiv:1905.11453}; \textit{R. L. Fernandes} and \textit{D. Iglesias Ponte}, Lett. Math. Phys. 90, No. 1--3, 137--159, (2009; Zbl 1183.53075)]. Indeed, a possible approach to studying the integrability of a Poisson structure arising from reduction of a Dirac structure consists in first understanding the integrability of the original Dirac structure and then checking whether one of its integrations gives rise by reduction to an integration of the quotient Poisson structure.
First, the paper addresses the more general question: \textit{``Given a surjective submersion \(q\colon S\to M\) and a Lie algebroid \(A\to M\), how is the integrability of \(A\) related to the integrability of the pullback Lie algebroid \(q^!A\to S\)?''} Clearly, the integrability of \(A\) implies the integrability of \(q^!A\) (cf. Corollary 1.9 in [\textit{P. J. Higgins} and \textit{K. Mackenzie}, J. Algebra 129, No. 1, 194--230 (1990; Zbl 0696.22007)]) but the converse does not hold (Example 3.8). Theorem 3.2 answers the previous question as follows: \(A\) is integrable if and only if \(q^!A\) admits a \textit{\(q\)-admissible integration}, i.e.,
\begin{itemize}
\item the pullback Lie algebroid \(q^!A\Rightarrow S\) is integrable and
\item the inclusion \(\ker Tq\to{}q^!A\) is integrable by a Lie groupoid morphism \(\Phi\colon S\times_MS\rightarrow G\), where \(S\times_MS\rightrightarrows S\) is the submersion groupoid.
\end{itemize}
Moreover, if \(q^!A\) admits a \(q\)-admissible integration as above, and \(R\) denotes the equivalence relation on \(G\) given, for all compatible \(g\in G\) and \(x,y\in S\times_MS\), by \[ g\sim\Phi(x)g\Phi(y), \] then Theorem 3.2 proves that the quotient \(G/R\) inherits a unique Lie groupoid structure over \(M\) such that the quotient map \(Q\colon G\rightarrow G/R\) is a Lie groupoid morphism. Additionally, the quotient Lie groupoid \(G/R\) integrates \(A\), i.e. \(\operatorname{Lie}(G/R)\cong A\).
The main result of the paper, Theorem 3.10, specializes Theorem 3.2 to the case of Poisson and Dirac structures. So, given a surjective submersion \(q\colon S\to M\), it characterizes the integrability of a Poisson structure \(\pi\) on \(M\) in terms of the pullback Dirac structure \(q^!(\operatorname{graph}(\pi))\) on \(S\). Specifically, it proves that the Poisson structure \(\pi\) is integrable if and only if the Dirac structure \(q^!(\operatorname{graph}(\pi))\) admits a \emph{\(q\)-admissible presymplectic integration} \((G,\omega)\), i.e.
\begin{itemize}
\item the Dirac structure \(q^!(\operatorname{graph}(\pi))\) structure admits a \(q\)-admissible integration \(G\), with associated Lie groupoid morphism \(\Phi\colon S\times_MS\to G\), and
\item \(G\) is endowed with a multiplicative presymplectic form \(\omega\) such that its infinitesimal counterpart is provided by \(q^!(\operatorname{graph}(\pi))\) and \(\Phi^\ast\omega=0\),
\end{itemize}
Moreover, Theorem 3.10 also proves that there is a (unique) multiplicative symplectic form \(\overline{\omega}\in\Omega^2(G/R)\) such that \(Q^\ast\overline{\omega}=\omega\).
As an immediate consequence of its main result the paper recovers well-known results about coisotropic reduction (Proposition 3.13) and the weak Morita invariance of integrations (Proposition 3.15). Additionally, the paper also applies Theorem 3.10 to get entirely new results like the integrability of two special classes of geometric structures, namely:
\begin{itemize}
\item Dirac homogeneous spaces (Theorem 4.19) and
\item Poisson homogeneous spaces of symplectic groupoids integrationg Poisson groups (Theorem 4.36).
\end{itemize}
Reviewer: Alfonso Giuseppe Tortorella (Porto)Ressayre's pairs in the Kähler settinghttps://zbmath.org/1496.530892022-11-17T18:59:28.764376Z"Paradan, Paul-Emile"https://zbmath.org/authors/?q=ai:paradan.paul-emileAs the title indicates, this paper develops an analogue in Kähler geometry of a theory in algebraic geometry due to \textit{N. Ressayre} [Invent. Math. 180, No. 2, 389--441 (2010; Zbl 1197.14051)]. More precisely, given a connected reductive group \(G\) acting on a normal projective variety \(X\), \textit{Ressayre} has shown that the facets of particular polyhedral cones in \(\operatorname{Pic}^G(X)\) can be parametrised by (in)equalities satisfied by so-called well-covering pairs. In the present work, analogous notions of \emph{Ressayre's pairs} are introduced to parametrise the facets of the Kirwan polytope, which is defined using a Hamiltonian action of a connected compact Lie group \(K\) on a Kähler manifold \(M\). (To build the analogy, one uses the holomorphic action of the complexification \(K_{\mathbb{C}}\) on \(M\) seen as a complex manifold.) The author's main idea behind this parametrisation consists in considering several embeddings of polytopes in order to prove that they coincide and that they are nothing else than the Kirwan polytope. Interestingly, these (\emph{a priori} different) polytopes are defined from the same set of inequalities which depend on various notions of Ressayre's pairs, see the beginning of Section 4. Two specific examples of this parametrisation are worked out in Section 6. They concern actions of \(\tilde{K}_{\mathbb{C}}\times K_{\mathbb{C}}\) on \(\tilde{K}_{\mathbb{C}}\) and on \(\tilde{K}_{\mathbb{C}} \times V\), where \(K\hookrightarrow \tilde{K}\) is a closed connected Lie subgroup and \(V\) is a vector space with \(K_{\mathbb{C}}\)-linear action. The introduction is particularly well written and it gives a very precise outline of the objects at stake in this paper.
Reviewer: Maxime Fairon (Glasgow)On symplectic transformationshttps://zbmath.org/1496.530902022-11-17T18:59:28.764376Z"Springer, T. A."https://zbmath.org/authors/?q=ai:springer.tonny-albert.1Summary: This is an English translation of the Ph.D. thesis `Over symplectische transformaties' that Tonny Albert Springer, `born in's-Gravenhage in 1926', submitted as thesis for -- as is stated on the original frontispiece - \textit{the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m.}, with promotor Prof. dr. H. D. Kloosterman.Symplectic Banach-Mazur distances between subsets of \(\mathbb{C}^n\)https://zbmath.org/1496.530912022-11-17T18:59:28.764376Z"Usher, Michael"https://zbmath.org/authors/?q=ai:usher.michaelThis paper introduces and studies three Banach-Mazur type distance functions on the space of open Liouville domains.
A Liouville domain is a manifold \(U\) with boundary \(\partial U\) and a \(1\)-form \(\lambda\) such that \(d \lambda\) is a symplectic form on \(U\) and the Liouville vector field \(X\) of \(\lambda\) (denoted by \(\mathcal{L}_\lambda\) in the paper), characterized by \(\iota_X d\lambda =\lambda\), points outward along \(\partial U\). The main examples in the paper are star-shaped domains in \(\mathbb{C}^n\) with the standard Liouville form \(\lambda_0=\frac{1}{2}\sum x_i dy_i -y_i dx_i\). Due to certain difficulties in working with manifolds with boundaries, the paper defines and works with open Liouvilles domains; see Definition 1.1. Nevertheless, it is proved in Proposition 2.2 that every open Liouville can be approximated by Liouville domains inside, allowing the author to go back and forth between the two concepts depending on the context.
The three (distance) functions between two open Liouville domains \((U,\lambda)\) and \((V,\mu)\) are denoted by \(d_c\), \(\delta_f\), \(d_f\). By Definition 1.2:
(1) the coarse symmetric function \(d_c\) measures the infimum of the shrinking (with respect to the negative flow of \(X\)) needed to embed \((U,\lambda)\) into \((V,\mu)\) and vice versa;
(2) the non-symmetric Banach-Mazur type hemi-distance function \(\delta_f\) has an extra condition that requires the image of shrunk \(U\) in \(V\) to include a slightly more shrunk copy of \(V\);
(3) and finally, the fine distance function \(d_f\) is the symmetrization of \(\delta_f\).
After taking logarithm, all these functions satisfy the triangle inequality.
The main results of the paper give explicit examples in \((\mathbb{C}^n,\lambda_0)\), with \(n>1\), that behave interestingly with respect to these distance functions. Theorem 1.1 shows how far the function \(\delta_f\) is from being symmetric. For any open ellipsoid \(U\), it constructs an explicit sequence of Liouville domains \((U_i)_{i=1}^\infty\) such that \(\lim_{i\to \infty} \delta_f(U_i,U)= 1\) while \(\lim_{i\to \infty} \delta_f(U,U_i)= \infty\). In addition, since \(d_c\leq \delta_f \leq d_f\), the result above shows that \(d_f\) is much finer than \(d_c\).
Theorem 1.1 shows that the \(\log d_c\) and \(\log d_f\) define significantly different topologies on the set of star-shaped domains in \(\mathbb{C}^n\). The second main result, Theorem 1.2, shows that with respect to the distance function \(d_f\), the space of star-shaped domains in \(\mathbb{C}^n\) (with \(n>1\)) is ``infinitely'' large in the sense that every \(\mathbb{R}^N\) quasi-isometrically embeds in that.
In addition to constructing explicit examples, by cutting out pieces from an ellipsoid in \(\mathbb{C}^n\), the other main ingredient of the proofs involves relating Filtered Equivariant Symplectic Homology (or FE-SH) and the hemi-distance function \(\delta_f\). In Section~3.2, the definition of Symplectic Homology is adapted to the category of open Liouville domains. Including one Liouville domain into another gives an exact symplectic cobordism of their boundaries and a map between the corresponding Symplectic Homology groups. Proposition 3.1 is the main result connecting \(\delta_f\) and the changes in FE-SH (seen as a persistence module). Given a Liouville domain, Symplectic Homology is a homology theory built on ``good'' Reeb orbits of the boundary contact manifold. For the truncated ellipsoids that appear in the proof of Theorem 1, these Reeb orbits are explicitly described and studied in Lemma 5.2, 5.4, etc.
Reviewer: Mohammad Farajzadeh Tehrani (Iowa City)Symplectic homology of convex domains and Clarke's dualityhttps://zbmath.org/1496.530922022-11-17T18:59:28.764376Z"Abbondandolo, Alberto"https://zbmath.org/authors/?q=ai:abbondandolo.alberto"Kang, Jungsoo"https://zbmath.org/authors/?q=ai:kang.jungsooAuthors' abstract: We prove that the Floer complex associated with a convex Hamiltonian function on \(\mathbb{R}^ {2n}\) is isomorphic to the Morse complex of Clarke's dual action functional associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.
Reviewer: Alexander Felshtyn (Szczecin)Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifoldshttps://zbmath.org/1496.530932022-11-17T18:59:28.764376Z"Tervil, Brian"https://zbmath.org/authors/?q=ai:tervil.brianA high driving force in symplectic topology is the celebrated Arnold conjecture concerning the number of fixed points of a Hamitonian symplectomorphism. The analogue of the Arnold conjecture in contact topology was introduced by \textit{S. Sandon} [Geom. Dedicata 165, 95--110 (2013; Zbl 1287.53067)] through the notion of \textit{translated point}. The main result of the paper under review is a version of Sandon's conjecture, namely
\textbf{Theorem 1.1.1.}: Let \((M, \omega,. \mathbb{T})\) be a closed monotone symplectic toric manifold with primitive symplectic form. Assume that it is different from \((\mathbb{C}P^{n-1}, \omega _{FS}, \mathbb{T}^n/S^1)\) and let \((V, \xi:=ker\ \alpha )\) be the prequantization space over \((M, \omega )\) with Euler class \(-[\omega ]\). Then any \(\phi \in \operatorname{Cont}_0(V, \xi )\) has at least \(N_M\) \(\alpha \)-translated points.
The minimal Chern number \(N_M\) of a monotone symplectic toric manifold is always strictly smaller than its cuplength \(cl(M)=\dim _{\mathbb{C}}M+1\) unless \((M, \omega )=(\mathbb{C}P^{n-1}, \omega _{FS})\) when both quantities equal \(n\).
Reviewer: Mircea Crâşmăreanu (Iaşi)An algebra of distributions related to a star product with separation of variableshttps://zbmath.org/1496.530942022-11-17T18:59:28.764376Z"Karabegov, Alexander"https://zbmath.org/authors/?q=ai:karabegov.alexander-vSummary: Given a star product with separation of variables \(\star\) on a pseudo-Kähler manifold \(M\) and a point \(x_0 \in M\), we construct an associative algebra of formal distributions supported at \(x_0\). We use this algebra to express the formal oscillatory exponents of a family of formal oscillatory integrals related to the star product \(\star \).Volume properties and rigidity on self-expanders of mean curvature flowhttps://zbmath.org/1496.530952022-11-17T18:59:28.764376Z"Ancari, Saul"https://zbmath.org/authors/?q=ai:ancari.saul"Cheng, Xu"https://zbmath.org/authors/?q=ai:cheng.xuSummary: In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the \(L\)-stability operator and discuss the \(L\)-stability of some special self-expanders. Besides, we prove that the surfaces \(\Gamma \times \mathbb{R}\) with the product metric are the only complete self-expander surfaces immersed in \(\mathbb{R}^3\) with constant scalar curvature, where \(\Gamma\) is a complete self-expander curve (properly) immersed in \(\mathbb{R}^2\).The level set flow of a hypersurface in \(\mathbb{R}^4\) of low entropy does not disconnecthttps://zbmath.org/1496.530962022-11-17T18:59:28.764376Z"Bernstein, Jacob"https://zbmath.org/authors/?q=ai:bernstein.jacob"Wang, Shengwen"https://zbmath.org/authors/?q=ai:wang.shengwenSummary: We show that if \(\Sigma \subset \mathbb{R}^4\) is a closed, connected hypersurface with entropy \(\lambda (\Sigma) \leq \lambda (\mathbb{S}^2 \times \mathbb{R})\), then the level set flow of \(\Sigma\) never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in \(\mathbb{R}^4\) of low entropy.Mean convex mean curvature flow with free boundaryhttps://zbmath.org/1496.530972022-11-17T18:59:28.764376Z"Edelen, Nick"https://zbmath.org/authors/?q=ai:edelen.nick"Haslhofer, Robert"https://zbmath.org/authors/?q=ai:haslhofer.robert"Ivaki, Mohammad N."https://zbmath.org/authors/?q=ai:ivaki.mohammad-n"Zhu, Jonathan J."https://zbmath.org/authors/?q=ai:zhu.jonathan-j-hSummary: In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [\textit{B. White}, J. Am. Math. Soc. 13, No. 3, 665--695 (2000; Zbl 0961.53039); J. Am. Math. Soc. 16, No. 1, 123--138 (2003; Zbl 1027.53078); Calc. Var. Partial Differ. Equ. 54, No. 2, 1457--1468 (2015; Zbl 1325.53090)] to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple-approximation scheme, which combines ideas from \textit{N. Edelen} [Adv. Math. 294, 1--36 (2016; Zbl 1341.35176)],
\textit{R. Haslhofer} and \textit{O. Hershkovits} [Adv. Math. 329, 1137--1155 (2018; Zbl 1387.53052)], and
\textit{A. Volkmann} [Free boundary problems governed by mean curvature. Freie Universität Berlin (PhD Thesis) (2015)]. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low-entropy free boundary flows in a half-slab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean-convex domains, as possible limit flows.Long time behavior of discrete volume preserving mean curvature flowshttps://zbmath.org/1496.530982022-11-17T18:59:28.764376Z"Morini, Massimiliano"https://zbmath.org/authors/?q=ai:morini.massimiliano"Ponsiglione, Marcello"https://zbmath.org/authors/?q=ai:ponsiglione.marcello"Spadaro, Emanuele"https://zbmath.org/authors/?q=ai:spadaro.emanuele-nunzioSummary: In this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.Convergence result and blow-up examples for the Guan-Li mean curvature flow on warped product spaceshttps://zbmath.org/1496.530992022-11-17T18:59:28.764376Z"Vétois, Jérôme"https://zbmath.org/authors/?q=ai:vetois.jeromeSummary: We examine the question of convergence of solutions to a geometric flow which was introduced by \textit{P. Guan} and \textit{J. Li} [Int. Math. Res. Not. 2015, No. 13, 4716--4740 (2015; Zbl 1342.53090)] for starshaped hypersurfaces in space forms and generalized by \textit{P. Guan} et al. [Trans. Am. Math. Soc. 372, No. 4, 2777--2798 (2019; Zbl 1421.53067)] to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.Elliptic gradient estimates for a nonlinear \(f\)-heat equation on weighted manifolds with evolving metrics and potentialshttps://zbmath.org/1496.531002022-11-17T18:59:28.764376Z"Abolarinwa, Abimbola"https://zbmath.org/authors/?q=ai:abolarinwa.abimbola"Taheri, Ali"https://zbmath.org/authors/?q=ai:taheri.ali|taheri.ali-karimiSummary: We develop local elliptic gradient estimates for a basic nonlinear \(f\)-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed.On the regularity of Ricci flows coming out of metric spaceshttps://zbmath.org/1496.531012022-11-17T18:59:28.764376Z"Deruelle, Alix"https://zbmath.org/authors/?q=ai:deruelle.alix"Schulze, Felix"https://zbmath.org/authors/?q=ai:schulze.felix"Simon, Miles"https://zbmath.org/authors/?q=ai:simon.milesThe authors investigate the following problem: consider a possibly incomplete Ricci flow \((M,g(t))_{t \in (0,T)}\) satisfying
\[ |\mathrm{Rm}|\leq c_0/t \tag{1} \]
and whose associated metric spaces \((M,d_{g(t)})\) Gromov-Hausdorff converge to a metric space \((X,d_X)\) as \(t \searrow 0\). What additional assumptions on \((X,d_X)\) and \((M,g(t))_{t \in (0,T)}\) guarentee that \(g(t)\) converges locally smoothly (or continuously) to a smooth (or continous) metric as \(t \searrow 0\)?
The authors give a positive answer under the additional assumption that for all \(t \in (0,T)\) \[ \mathrm{Ric}_{g(t)}\geq -1, \tag{2}\] \(B_{g(t)}(x_0,1) \Subset M\) and that \((X,d_X)\) is \emph{smoothly (or continously) \(n\)-Riemannian}. One says that \((X,d_X)\) is \emph{smoothly (respectively continously) \(n\)-Riemannian} if for any \(x_0 \in X\) there are \(0<\tilde r<r\) with \(\tilde r<r/5\) and points \(a_1,\ldots,a_n\in B_{d_X}(x_0,r)\) such that the map \[F(x) = (d_X(a_1,x),\ldots,d_X(a_n,x)),\quad x \in B_{d_X}(x_0,r)\] is an \((1+\varepsilon_0)\)-bilipschitz homeomorphism on \(B_{d_X}(x_0,5\tilde r)\), and the pushed forward of \(d_X\) via \(F\) is on \(\mathbb{B}_{4\tilde r}(F(x_0))\Subset F(B_{d_X}(x_0,5\tilde r))\) induced by a smooth (respectively continous) Riemannian metric. The smooth definition corresponds to \((X,d_X)\) being locally isometric to smooth \(n\)-Riemannian manifolds.
Before stating the main results let us recall a bit of context. When all \(g(t)\) are complete and satisfy (1) and (2), \textit{M. Simon} and \textit{P. M. Topping} [Geom. Topol. 25, No. 2, 913--948 (2021; Zbl 1470.53083)] show that \(d_{g(t)} \to d_0\) a metric on \(M\) as \(t \searrow 0\), and that \((M,d_0)\) is isometric to \((X,d_X)\). Without the completeness assumption, a localized version of this holds assuming that \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\). Then \(X:=\cap_{s \in (0,T)} B_{g(s)}(x_0,1/2)\) is non empty and is endowed with a well defined limiting metric \(d_0=\lim d_{g(t)}\) as \(t \searrow 0\). Moreover \(B_{g(t)}(x_0,r) \Subset \mathcal{X} \subset X\) for all \(r \leq R(c_0,n)\) and \(t \leq S(c_0,n)\), where \(\mathcal{X}\) is the connected component of \(X\) containing \(x_0\), and the topology of \(B_{d_0}(x_0,r)\) induced by \(d_0\) agrees with that of \(M\).
The main results of the authors are the following.
Theorem 1.6 : Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is smoothly \(n\)-Riemannian for some \(r<R(c_0,n)\). Then there exists a smooth Riemannian metric \(g_0\) on \(B_{d_0}(x_0,s)\) for some \(s<r\) such that \(g(t)\) extends to a smooth solution \((B_{d_0}(x_0,s), g(t))_{t \in [0,T)}\) by defining \(g(0)=g_0\).
Theorem 1.7: Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is continuously \(n\)-Riemannian for some \(r<R(c_0,n)\). Then for any stricty monotone sequence \(t_i \searrow 0\), there exists \(v>0\) and a continuous Riemannian metric \(\tilde g_0\) defined on \(\mathbb{B}_v(p) \subset \mathbb{R}^n\) and a family of smooth diffeomorphisms \(Z_i : B_{d_0}(x_0,2v) \to \mathbb{R}^n\) such that \((Z_i)_\ast(g(t_i))\) converges in the \(C^0\)-sense to \(\tilde g_0\) as \(t_i \searrow\) on \(\mathbb{B}_v(p)\).
Reviewer: Laurent Bessières (Bordeaux)Nguyen's tridents and the classification of semigraphical translators for mean curvature flowhttps://zbmath.org/1496.531022022-11-17T18:59:28.764376Z"Hoffman, David"https://zbmath.org/authors/?q=ai:hoffman.david-a"Martín, Francisco"https://zbmath.org/authors/?q=ai:martin.francisco"White, Brian"https://zbmath.org/authors/?q=ai:white.brian-cabellSummary: We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to 0 to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.Variational principles and combinatorial \(p\)-th Yamabe flows on surfaceshttps://zbmath.org/1496.531032022-11-17T18:59:28.764376Z"Li, Chunyan"https://zbmath.org/authors/?q=ai:li.chunyan"Lin, Aijin"https://zbmath.org/authors/?q=ai:lin.aijin"Yang, Chang"https://zbmath.org/authors/?q=ai:yang.changThe authors start giving details on PL-metrics (piecewise linear metrics) in Euclidean triangulated manifolds. For these types of metrics, the most natural curvature is the combinatorial Gauss curvature. Considering the combinatorial Gauss-Bonnet formula associated with this curvature one obtains the average of the total combinatorial Gauss curvature designated by \(K_{av}\). The constant combinatorial PL-metric has curvature \(K_{av}\) at all vertices. The Yamabe combinatorial problem deals with the existence of this metric.
The authors start by explaining previous studies on this problem done by other mathematicians. \textit{F. Luo} [Commun. Contemp. Math. 6, No. 5, 765--780 (2004; Zbl 1075.53063)] studied the discrete Yamabe problem through the combinatorial Yamabe flow. Luo showed that the combinatorial Yamabe flow is the negative gradient of a potential functional \(F\). He showed that \(F\) is locally convex and obtained the local rigidity for the curvature map and the local convergence of the combinatorial Yamabe flow. He also conjectured that the combinatorial Yamabe flow converges to a constant curvature PL-metric after a finite number of surgeries on the triangulation. \textit{H. Ge} and \textit{W. Jiang} [Calc. Var. Partial Differ. Equ. 55, No. 6, Paper No. 136, 14 p. (2016; Zbl 1359.53054)] introduced the extended combinatorial Yamabe algorithm to handle possible singularities along the combinatorial Yamabe flow and \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] started doing surgery by ``flipping the algorithm''.
In this paper the authors generalize results for both extended combinatorial Yamabe flow and combinatorial Yamabe flow with surgery. The generalization is done first for \(p>1\) introducing the extended combinatorial \(p\)-th Yamabe flow, which is the extended Yamabe flow when \(p=2\) introduced by Ge and Jiang and then generalize the main results they obtained. They show that the solution to the extended combinatorial \(p\)-th Yamabe flow exists for all time. The details are all explained in Section 2. In Section 3 the authors recall the discrete conformal theory and discrete unifomization theorem established by \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] and generalize their results for the combinatorial \(p\)-th Yamabe flow with surgery. It is shown that for the generalized \(p\)-th flows \(p>1\) and \(p\neq 2\) there exists only curvature convergence but no exponential convergence as in the case of \(p=2\).
Reviewer: Ana Pereira do Vale (Braga)Examples of Ricci limit spaces with non-integer Hausdorff dimensionhttps://zbmath.org/1496.531042022-11-17T18:59:28.764376Z"Pan, Jiayin"https://zbmath.org/authors/?q=ai:pan.jiayin"Wei, Guofang"https://zbmath.org/authors/?q=ai:wei.guofangSummary: We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of \textit{J. Cheeger} and \textit{T. H. Colding} [J. Differ. Geom. 54, No. 1, 13--35 (2000; Zbl 1027.53042), 15 p.] about collapsing Ricci limit spaces.Curve shortening flow on Riemann surfaces with possible ambient conic singularitieshttps://zbmath.org/1496.531052022-11-17T18:59:28.764376Z"Ma, Biao"https://zbmath.org/authors/?q=ai:ma.biao.1Summary: In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles smaller than or equal to \(\pi\).Quintessential effects on quasiperiodic oscillations in \(4D\) Einstein-Gauss-Bonnet gravityhttps://zbmath.org/1496.531062022-11-17T18:59:28.764376Z"Rayimbaev, Javlon"https://zbmath.org/authors/?q=ai:rayimbaev.javlon"Tadjimuratov, Pulat"https://zbmath.org/authors/?q=ai:tadjimuratov.pulat"Ahmedov, Bobomurat"https://zbmath.org/authors/?q=ai:ahmedov.bobomurat-j"Palvanov, Satimbay"https://zbmath.org/authors/?q=ai:palvanov.satimbayThe authors focused their attention on discussing the dynamics of test particles in a quintessential black hole space-time with the help of Einstein-Gauss-Bonnet theory of gravity. On this purpose, firstly, they analyzed the main features of space-time geometry and possible values for the coupling and quintessential parameters. Subsequently, they investigated (i) the case when the quintessential black hole event horizon lies at 2M, (ii) scalar invariants of the selected spacetime model, (iii) the effects of the quintessential and the Gauss-Bonnet parameters on the effective potential for radial motion, specific energy and angular momentum of the particles corresponding to their circular orbits, (iv) the effects of the parameters on innermost stable circular orbit radius, (v) the relations between the quintessential and the Gauss-Bonnet parameters providing 6M innermost stable circular orbit radius and (vi) the cases with fundamental frequencies such as the Keplerian and harmonic oscillations.
Reviewer: Mustafa Salti (Mersin)Geometry and generalization: eigenvalues as predictors of where a network will fail to generalizehttps://zbmath.org/1496.531072022-11-17T18:59:28.764376Z"Agarwala, Susama"https://zbmath.org/authors/?q=ai:agarwala.susama"Dees, Ben"https://zbmath.org/authors/?q=ai:dees.ben-k"Gearheart, Andrew"https://zbmath.org/authors/?q=ai:gearheart.andrew"Lowman, Corey"https://zbmath.org/authors/?q=ai:lowman.corey(no abstract)From deformation theory of wheeled props to classification of Kontsevich formality mapshttps://zbmath.org/1496.550122022-11-17T18:59:28.764376Z"Andersson, Assar"https://zbmath.org/authors/?q=ai:andersson.assar"Merkulov, Sergei"https://zbmath.org/authors/?q=ai:merkulov.sergei-aAuthors' abstract: We study the homotopy theory of the wheeled prop controlling Poisson structures on formal graded finite-dimensional manifolds and prove, in particular, that the Grothendieck-Teichmüller group acts on that wheeled prop faithfully and homotopy nontrivially. Next, we apply this homotopy theory to the study of the deformation complex of an arbitrary Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by \textit{M. Kontsevich} [Math. Phys. Stud. 20, 139--156 (1997; Zbl 1149.53325)] and studied by \textit{T. Willwacher} [Invent. Math. 200, No. 3, 671--760 (2015; Zbl 1394.17044)].
Reviewer: Iakovos Androulidakis (Athína)A Möbius invariant discretization of O'Hara's Möbius energyhttps://zbmath.org/1496.570042022-11-17T18:59:28.764376Z"Blatt, Simon"https://zbmath.org/authors/?q=ai:blatt.simon|blatt.simon.1"Ishizeki, Aya"https://zbmath.org/authors/?q=ai:ishizeki.aya"Nagasawa, Takeyuki"https://zbmath.org/authors/?q=ai:nagasawa.takeyukiThe authors introduce a new discretization of O'Hara's Möbius energy. In contrast to the known discretizations of \textit{J. K. Simon} [J. Knot Theory Ramifications 3, No. 3, 299--320 (1994; Zbl 0841.57017)] and of \textit{D. Kim} and \textit{R. Kusner} [Exp. Math. 2, No. 1, 1--9 (1993; Zbl 0818.57007)] the new discretization is invariant under Möbius transformations of the surrounding space. Moreover, this energy is minimized by polygons with vertices on a circle. The starting point for this new discretization is the so-called cosine formula of Doyle and Schramm. In addition, the authors then show \(\Gamma\)-convergence of the discretized energy to the Möbius energy provided that the fineness of the polygons is going to 0. (Here a map \(p : \mathbb{R/Z}\to \mathbb{R}^n\) a closed polygon with the \(m\) vertices \(p(\theta_i)\in\mathbb{R}^n\), \(i=1,\ldots,m\) if there are points \(\theta_i \in[0,1)\), \(\theta_1 <\theta_2 <\dots<\theta_m\) such that \(p\) is linear between two neighboring points \(\theta_i\) and \(\theta_{i+1}\) with \(\theta_{m+1}=\theta_1\). The fineness is then defined as \(\max|\theta_{i+1}-\theta_i|\).)
Reviewer: Claus Ernst (Bowling Green)Contact geometry and the mapping class grouphttps://zbmath.org/1496.570212022-11-17T18:59:28.764376Z"Licata, Joan E."https://zbmath.org/authors/?q=ai:licata.joan-eThis is a very short exposition of cutting and gluing of 3-manifolds, mapping class groups, contact structures and their interrelations.Erratum to: ``A note on the McShane's identity for Hecke groups''https://zbmath.org/1496.570242022-11-17T18:59:28.764376Z"Farooq, K."https://zbmath.org/authors/?q=ai:farooq.kErratum to the author's paper [ibid. 52, No. 3, 915--931 (2021; Zbl 1489.57015)].Sheaves via augmentations of Legendrian surfaceshttps://zbmath.org/1496.570282022-11-17T18:59:28.764376Z"Rutherford, Dan"https://zbmath.org/authors/?q=ai:rutherford.dan"Sullivan, Michael"https://zbmath.org/authors/?q=ai:sullivan.michael-gAuthors' abstract: Given an augmentation for a Legendrian surface in a 1-jet space, \(\Lambda\subset J^1(M)\), we explicitly construct an object, \(F\in Sh_\Lambda(M\times R,K)\), of the (derived) category from [\textit{V. Shende} et al., Invent. Math. 207, No. 3, 1031--1133 (2017; Zbl 1369.57016)] of constructible sheaves on \(M\times \mathbb R\) with singular support determined by \(\Lambda\). In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on [\textit{D. Rutherford} and \textit{M. Sullivan}, Adv. Math. 374, Article ID 107348, 71 p. (2020; Zbl 1475.53099); Int. J. Math. 30, No. 7, Article ID 1950036, 135 p. (2019; Zbl 1420.53088); ibid. 30, No. 7, Article ID 1950037, 111 p. (2019; Zbl 1419.53079)], is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of Shende et al. [loc. cit.] for 1-dimensional Legendrian knots to obtain a combinatorial model for sheaves in \( Sh_\Lambda(M\times \mathbb R,K)\) in the 2-dimensional case.
Reviewer: Alexander Felshtyn (Szczecin)Short survey on the existence of slices for the space of Riemannian metricshttps://zbmath.org/1496.580022022-11-17T18:59:28.764376Z"Corro, Diego"https://zbmath.org/authors/?q=ai:corro.diego"Kordaß, Jan-Bernhard"https://zbmath.org/authors/?q=ai:kordass.jan-bernhardThe article in question gives a survey about slice theorems and an alternative proof of Ebin's slice theorem. The latter roughly states, that any Riemannian metric \(g\) on a given manifold \(M\) admits an open neighbourhood diffeomorphic to \(S_g\times(\mathrm{Diff(M)}/{\mathrm{Diff}(M)_g})\), where \(\mathrm{Diff}(M)\) is the group of self-diffeomorphisms of \(M\), \(\mathrm{Diff}(M)_g\) is the subgroup of isometries of \(g\) and \(S_g\) is a certain submanifold of the space of all Riemannian metrics called the slice. In contrast to the original proof of \textit{D. G. Ebin} [Proc. Sympos. Pure Math. 15, 11--40 (1970; Zbl 0205.53702)], the authors avoid technical work done in the context of Sobolev spaces. They are also able to show the existence of an equivariant tubular neighbourhood of the orbit \(\mathrm{Diff}(M)\cdot g\) which is homeomorphic to \(\mathrm{Diff}(M)\times_{\mathrm{Diff}(M)_g} S_g\).
For the entire collection see [Zbl 1495.53005].
Reviewer: Georg Frenck (Augsburg)The derivatives of the heat kernel on complete manifoldshttps://zbmath.org/1496.580072022-11-17T18:59:28.764376Z"Fotiadis, Anestis"https://zbmath.org/authors/?q=ai:fotiadis.anestisThe author uses a new iteration argument to obtain the estimates for the time derivatives of the heat kernel on complete non-compact manifolds. Then the author applies these estimates to study the \(L^p\)-boundedness of the Littlewood-Paley-Stein operators on a class of locally symmetric spaces.
Reviewer: Shu-Yu Hsu (Chiayi)Resolvent estimates on asymptotically cylindrical manifolds and on the half linehttps://zbmath.org/1496.580102022-11-17T18:59:28.764376Z"Christiansen, Tanya J."https://zbmath.org/authors/?q=ai:christiansen.tanya-j"Datchev, Kiril"https://zbmath.org/authors/?q=ai:datchev.kiril-rIn this article the authors study the spectral and scattering theory for a class of asympotically cylindrical manifolds with sufficiently mild geodesic trapping. One example of such a manifold is the cigar-shaped warped product (\(\mathbb{R}^d\), \(g_0\)), \(d \ge 2\), with metric \(g_0 = dr^2 + F(r) dS\), where \(r\) is the radial variable, \(dS\) is the usual metric on the unit sphere, and \(F(r) = r^2\) near \(r = 0\), while \(F'\) is compactly supported on some interval \([0,R]\) and positive on \((0,R)\). Another example is a convex cocompact hyperbolic surface \((X,g_H)\) for which there is a compact set \(N \subseteq X\) such that
\[
X \setminus N = (0,\infty)_r \times Y_y, \quad g_H\rvert_{X \setminus N} = dr^2 + \cosh^2 r dy^2,
\]
where \(Y\) is a disjoint union of \(k \ge 1\) geodesic circles. Indeed, in the first example, the only trapped geodesics are the circular ones on the cylindrical end (and this is the smallest amount of trapping a manifold with a cylindrical end can have.)
Suppose that trapping is suitably mild, in the sense that, in the presence of complex absorption, the resolvent is bounded polynomially in the spectral parameter \(z\) as \(\text{Re}\, z \to \infty\). Then the authors show that the number of embedded resonances and eigenvalues is finite, and that the cutoff resolvent (without complex absoprtion) is uniformly bounded as \(\text{Re}\, z \to \infty\). This bound is sharp in the setting of the first example described above.
Along the way to their main result, the authors also prove some resolvent estimates for repulsive potentials on the half-line.
Reviewer: Jacob Shapiro (Dayton)Evolving finite element methods with an artificial tangential velocity for mean curvature flow and Willmore flowhttps://zbmath.org/1496.651632022-11-17T18:59:28.764376Z"Hu, Jiashun"https://zbmath.org/authors/?q=ai:hu.jiashun"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyangSummary: An artificial tangential velocity is introduced into the evolving finite element methods for mean curvature flow and Willmore flow proposed by \textit{B. Kovács} et al. [Numer. Math. 143, No. 4, 797--853 (2019; Zbl 1427.65250); ibid. 149, No. 3, 595--643 (2021; Zbl 1496.65165)] in order to improve the mesh quality in the computation. The artificial tangential velocity is constructed by considering a limiting situation in the method proposed by \textit{J. W. Barrett} et al. [J. Comput. Phys. 222, No. 1, 441--467 (2007; Zbl 1112.65093); ibid. 227, No. 9, 4281--4307 (2008; Zbl 1145.65068); SIAM J. Sci. Comput. 31, No. 1, 225--253 (2008; Zbl 1186.65133)]. The stability of the artificial tangential velocity is proved. The optimal-order convergence of the evolving finite element methods with artificial tangential velocity are proved for both mean curvature flow and Willmore flow. Extensive numerical experiments are presented to illustrate the convergence of the method and the performance of the artificial tangential velocity in improving the mesh quality.Approximate Petz recovery from the geometry of density operatorshttps://zbmath.org/1496.810282022-11-17T18:59:28.764376Z"Cree, Sam"https://zbmath.org/authors/?q=ai:cree.sam"Sorce, Jonathan"https://zbmath.org/authors/?q=ai:sorce.jonathanSummary: We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy \(\widetilde{D}_2 \). For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-\(\epsilon\) accuracy whenever the data processing inequality for \(\widetilde{D}_2\) is saturated up to terms of order \(\epsilon^2\) times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending [\textit{S. Cree} and \textit{J. Sorce}, ``Geometric conditions for saturating the data processing inequality'', J. Phys. A, Math. Theor. 55, No. 13, Article ID 135202, 24 p. (2022; \url{doi:10.1088/1751-8121/ac5648})], in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched Rényi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched Rényi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.Construction of quantum target space from world-sheet states using quantum state tomographyhttps://zbmath.org/1496.810332022-11-17T18:59:28.764376Z"Wani, Salman Sajad"https://zbmath.org/authors/?q=ai:wani.salman-sajad"Shabir, Arshid"https://zbmath.org/authors/?q=ai:shabir.arshid"Hassan, Junaid Ul"https://zbmath.org/authors/?q=ai:hassan.junaid-ul"Kannan, S."https://zbmath.org/authors/?q=ai:kannan.senthamarai|kannan.sriraman|kannan.srinathan|kannan.sriram|kannan.siddarth|kannan.sreeram"Patel, Hrishikesh"https://zbmath.org/authors/?q=ai:patel.hrishikesh"Sudheesh, C."https://zbmath.org/authors/?q=ai:sudheesh.c"Faizal, Mir"https://zbmath.org/authors/?q=ai:faizal.mirSummary: In this paper, we will construct the quantum states of target space coordinates from world-sheet states, using quantum state tomography. To perform quantum state tomography of an open string, we will construct suitable quadrature operators. We do this by first defining the quadrature operators in world-sheet, and then using them to construct the quantum target space quadrature operators for an open string. We will connect the quantum target space to classical geometry using coherent string states. We will be using a novel construction based on a string displacement operator to construct these coherent states. The coherent states of the world-sheet will also be used to construct the coherent states in target space.(2+1)-dimensional unstable matter waves in self-interacting Pseudospin-1/2 BECs under combined Rashba and Dresselhaus spin-orbit couplingshttps://zbmath.org/1496.811092022-11-17T18:59:28.764376Z"Tabi, Conrad Bertrand"https://zbmath.org/authors/?q=ai:tabi.conrad-bertrand"Veni, Saravana"https://zbmath.org/authors/?q=ai:veni.saravana"Kofané, Timoléon Crépin"https://zbmath.org/authors/?q=ai:kofane.timoleon-crepinSummary: The modulational instability (MI) of continuous waves is exclusively addressed theoretically and numerically in a two-component Bose-Einstein condensate in the presence of a mixture of Rashba and Dresselhaus (RD) spin-orbit couplings and the Lee-Huang-Yang (LHY) term. The linear stability analysis is utilized to derive an expression for the MI growth rate. It is revealed that instability can be excited in the presence of the RD spin-orbit coupling under conditions where nonlinear and dispersive effects are suitably balanced. Analytical predictions are confirmed via direct numerical simulations, where MI is manifested by the emergence of soliton-molecules that include four-peaked solitons and more exotic vortex structures that are very sensitive to variations in spin-orbit coupling strengths. Our study suggests that MI is a suitable mechanism for generating matter waves through multi-peaked solitons of various geometries.Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-timeshttps://zbmath.org/1496.830022022-11-17T18:59:28.764376Z"Gottschalk, Hanno"https://zbmath.org/authors/?q=ai:gottschalk.hanno"Rothe, Nicolai R."https://zbmath.org/authors/?q=ai:rothe.nicolai-r"Siemssen, Daniel"https://zbmath.org/authors/?q=ai:siemssen.danielThe Einstein equations and multipole moments at null infinityhttps://zbmath.org/1496.830032022-11-17T18:59:28.764376Z"Tafel, J."https://zbmath.org/authors/?q=ai:tafel.jacekJacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effecthttps://zbmath.org/1496.830042022-11-17T18:59:28.764376Z"Chanda, Sumanto"https://zbmath.org/authors/?q=ai:chanda.sumanto"Gibbons, G. W."https://zbmath.org/authors/?q=ai:gibbons.gary-william"Guha, Partha"https://zbmath.org/authors/?q=ai:guha.partha"Maraner, Paolo"https://zbmath.org/authors/?q=ai:maraner.paolo"Werner, Marcus C."https://zbmath.org/authors/?q=ai:werner.marcus-cSummary: In this paper, we return to the subject of Jacobi metrics for timelike and null geodesics in stationary spacetimes, correcting some previous misconceptions. We show that not only null geodesics but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulas. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally, we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behavior of Maxwell's equations in the curved spacetime. In particular, we see in very concrete terms how the gravitational electric permittivity, magnetic permeability, and magnetoelectric susceptibility enter the Jacobi-Maupertuis Randers-Finsler function.\par{\copyright 2019 American Institute of Physics}Möbius mirrorshttps://zbmath.org/1496.830062022-11-17T18:59:28.764376Z"Good, Michael R. R."https://zbmath.org/authors/?q=ai:good.michael-r-r"Linder, Eric V."https://zbmath.org/authors/?q=ai:linder.eric-vOn almost Ehlers-Geren-Sachs theoremshttps://zbmath.org/1496.830072022-11-17T18:59:28.764376Z"Lee, Ho"https://zbmath.org/authors/?q=ai:lee.ho"Nungesser, Ernesto"https://zbmath.org/authors/?q=ai:nungesser.ernesto"Stalker, John"https://zbmath.org/authors/?q=ai:stalker.john-gConformally flat pseudoprojective symmetric spacetimes in \(f(R, \mathcal{G})\) gravityhttps://zbmath.org/1496.830102022-11-17T18:59:28.764376Z"De, Uday Chand"https://zbmath.org/authors/?q=ai:de.uday-chand"Shenawy, Sameh"https://zbmath.org/authors/?q=ai:shenawy.sameh"Syied, Abdallah Abdelhameed"https://zbmath.org/authors/?q=ai:syied.abdallah-abdelhameed"Turki, Nasser Bin"https://zbmath.org/authors/?q=ai:turki.nasser-bin(no abstract)Darboux vector in four-dimensional space-timehttps://zbmath.org/1496.830112022-11-17T18:59:28.764376Z"Hu, Na"https://zbmath.org/authors/?q=ai:hu.na"Zhang, Tingting"https://zbmath.org/authors/?q=ai:zhang.tingting"Jiang, Yang"https://zbmath.org/authors/?q=ai:jiang.yang(no abstract)Shadows in conformally related gravity theorieshttps://zbmath.org/1496.830132022-11-17T18:59:28.764376Z"Pal, Kunal"https://zbmath.org/authors/?q=ai:pal.kunal"Pal, Kuntal"https://zbmath.org/authors/?q=ai:pal.kuntal"Shaikh, Rajibul"https://zbmath.org/authors/?q=ai:shaikh.rajibul"Sarkar, Tapobrata"https://zbmath.org/authors/?q=ai:sarkar.tapobrataSummary: Null geodesics are invariant under a conformal transformation, and thus it might seem natural to assume that the observables corresponding to the shadow of a space-time are also conformally invariant. Here, we argue instead, that since the Arnowitt-Deser-Misner mass and the active gravitational mass of an asymptotically flat space-time are not, in general, invariant under such conformal transformations, the shadow radius for photon motion in a space-time would be quantitatively different, when viewed from two different conformally related frames, although the expression for the shadow radius is similar. We then use this fact to propose a novel method to constrain the relevant parameters in a gravity theory conformally related to general relativity. As examples of our method, we constrain the parameter space in Brans-Dicke theory, and a class of brane-world gravity models, by using the recent observational data of M\(87^\ast\) by the Event Horizon Telescope.Generalized geodesic deviation in de Sitter spacetimehttps://zbmath.org/1496.830142022-11-17T18:59:28.764376Z"Waldstein, Isaac Raj"https://zbmath.org/authors/?q=ai:waldstein.isaac-raj"Brown, J. David"https://zbmath.org/authors/?q=ai:brown.j-davidLocal conformal instability and local non-collapsing in the Ricci flow of quantum spacetimehttps://zbmath.org/1496.830172022-11-17T18:59:28.764376Z"Luo, M. J."https://zbmath.org/authors/?q=ai:luo.meiju|luo.meijin|luo.ming-jianSummary: It is known that the conformal instability or bottomless problem rises in the path integral method in quantizing the general relativity. Does quantum spacetime itself really suffer from such conformal instability? If so, does the conformal instability cause the collapse of local spacetime region or even collapse the whole spacetime? The problems are studied in the framework of the Quantum Spacetime Reference Frame (QSRF) and induced spacetime Ricci flow. We find that if the lowest eigenvalue of an operator, associated with the F-functional in a local compact (closed and bounded) region, is positive, the local region is conformally unstable and will tend to volume-shrinking and curvature-pinching along the Ricci flow-time t; if the eigenvalue is negative or zero, the local region is conformally stable up to a trivial rescaling. However, the local non-collapsing theorem in the Ricci flow proved by Perelman ensures that the instability will not cause the local compact spacetime region collapse into nothing. The total effective action is also proved positive defined and bounded from below keeping the whole spacetime conformally stable, which can be considered as a generalization of the classical positive mass theorem of gravitation to the quantum level.An effective model for the quantum Schwarzschild black holehttps://zbmath.org/1496.830202022-11-17T18:59:28.764376Z"Alonso-Bardaji, Asier"https://zbmath.org/authors/?q=ai:alonso-bardaji.asier"Brizuela, David"https://zbmath.org/authors/?q=ai:brizuela.david"Vera, Raül"https://zbmath.org/authors/?q=ai:vera.raulSummary: We present an effective theory to describe the quantization of spherically symmetric vacuum motivated by loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation and a linear combination of constraints of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambiguous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model.Aspects of three-dimensional higher curvatures gravitieshttps://zbmath.org/1496.830252022-11-17T18:59:28.764376Z"Bueno, Pablo"https://zbmath.org/authors/?q=ai:bueno.pablo"Cano, Pablo A."https://zbmath.org/authors/?q=ai:cano.pablo-a"Llorens, Quim"https://zbmath.org/authors/?q=ai:llorens.quim"Moreno, Javier"https://zbmath.org/authors/?q=ai:moreno.javier"van der Velde, Guido"https://zbmath.org/authors/?q=ai:van-der-velde.guidoTopological confinement in Skyrme holographyhttps://zbmath.org/1496.830262022-11-17T18:59:28.764376Z"Cartwright, Casey"https://zbmath.org/authors/?q=ai:cartwright.casey"Harms, Benjamin"https://zbmath.org/authors/?q=ai:harms.benjamin-c"Kaminski, Matthias"https://zbmath.org/authors/?q=ai:kaminski.matthias"Thomale, Ronny"https://zbmath.org/authors/?q=ai:thomale.ronnyAdS-dS stationary rotating black hole exact solution within Einstein-nonlinear electrodynamicshttps://zbmath.org/1496.830282022-11-17T18:59:28.764376Z"García-Díaz, Alberto A."https://zbmath.org/authors/?q=ai:garcia-diaz.alberto-aSummary: In this report the exact rotating charged black hole solution to the Einstein-nonlinear electrodynamics theory with a cosmological constant is presented. This black hole is equipped with mass, rotation parameter, electric and magnetic charges, cosmological constant \(\Lambda\), and three parameters due to the nonlinear electrodynamics: \(\beta\) is associated to the potential vectors \(A_\mu\) and \(^\star P_\mu\), and two constants, \(F_0\) and \(G_0\), due to the presence of the invariants \(F\) and \(G\) in the Lagrangian \(L(F(x^\mu),G(x^\mu))\). This solution is of Petrov type D, characterized by the Weyl tensor eigenvalue \(\Psi_2\), the traceless Ricci tensor eigenvalue \(S=2\Phi_{(11)}\), and the scalar curvature \(R\); it allows for event horizons, exhibits a ring singularity and fulfils the energy conditions. Its Maxwell limit is the de Sitter-Anti-de Sitter-Kerr-Newman black hole solution.The Penrose property with a cosmological constanthttps://zbmath.org/1496.830312022-11-17T18:59:28.764376Z"Cameron, Peter"https://zbmath.org/authors/?q=ai:cameron.peter-jLaplacian on fuzzy de Sitter spacehttps://zbmath.org/1496.830332022-11-17T18:59:28.764376Z"Brkić, Bojana"https://zbmath.org/authors/?q=ai:brkic.bojana"Burić, Maja"https://zbmath.org/authors/?q=ai:buric.maja"Latas, Duško"https://zbmath.org/authors/?q=ai:latas.duskoInteraction of inhomogeneous axions with magnetic fields in the early universehttps://zbmath.org/1496.830382022-11-17T18:59:28.764376Z"Dvornikov, Maxim"https://zbmath.org/authors/?q=ai:dvornikov.maximSummary: We study the system of interacting axions and magnetic fields in the early universe after the quantum chromodynamics phase transition, when axions acquire masses. Both axions and magnetic fields are supposed to be spatially inhomogeneous. We derive the equations for the spatial spectra of these fields, which depend on conformal time. In case of the magnetic field, we deal with the spectra of the energy density and the magnetic helicity density. The evolution equations are obtained in the closed form within the mean field approximation. We choose the parameters of the system and the initial condition which correspond to realistic primordial magnetic fields and axions. The system of equations for the spectra is solved numerically. We compare the cases of inhomogeneous and homogeneous axions. The evolution of the magnetic field in these cases is different only within small time intervals. Generally, magnetic fields are driven mainly by the magnetic diffusion. We find that the magnetic field instability takes place for the amplified initial wavefunction of the homogeneous axion. This instability is suppressed if we account for the inhomogeneity of the axion.On maximal acceleration, strings with dynamical tension, and Rindler worldsheetshttps://zbmath.org/1496.830412022-11-17T18:59:28.764376Z"Castro Perelman, Carlos"https://zbmath.org/authors/?q=ai:castro-perelman.carlosSummary: Starting with a different action and following a different procedure than the construction of strings with dynamical tensions described by \textit{E. I. Guendelman} [``Implications of the spectrum of dynamically generated string tension theories'', Int. J. Mod. Phys. D 30, No.14, Article ID 2142028, 9 p. (2021; \url{doi:10.1142/S0218271821420281})], a variational procedure of our action leads to a coupled nonlinear system of \(D + 4\) partial differential equations for the \(D\) string coordinates \(X^\mu\) and the \textit{quartet} of scalar fields \(\varphi^1\), \(\varphi^2\), \(\phi\), \(T\), including the dilaton \(\phi(\sigma)\) and the tension \(T(\sigma)\) field. Trivial solutions to this system of complicated equations lead to a constant tension and to the standard string equations of motion. One of the most relevant features of our findings is that the Weyl invariance of the traditional Polyakov string is traded for the invariance under area-preserving diffeomorphisms. The final section is devoted to the physics of maximal proper forces (acceleration), minimal length within the context of Born's Reciprocal Relativity theory [\textit{C. Castro Perelman}, ``Is dark matter and black hole cosmology an effect of Born's reciprocal relativity theory?'', Can. J. Phys. 97, No. 2, 198--209 (2019; \url{doi.org/10.1139/cjp-2018-0097})] and to the Rindler world sheet description of accelerated \textit{open} and \textit{closed} strings from a very different approach and perspective than the one undertaken by [\textit{A. Bagchi} et al., ``A Rindler road to Carrollian worldsheets'', Preprint, \url{arXiv:2111.01172}; \textit{A. Bagchi}, \textit{A. Banerjee} and \textit{S. Chakrabotty}, ``Rindler physics on the string worldsheet'', Phys. Rev. Lett. 126, No. 3, Article ID 031601, 6 p. (2021; \url{doi:10.1103/PhysRevLett.126.031601})].On a class of orientable and nonorientable strip surfaceshttps://zbmath.org/1496.970072022-11-17T18:59:28.764376Z"Hernández, José Alberto Murillo"https://zbmath.org/authors/?q=ai:murillo-hernandez.jose-alberto(no abstract)