48,717 papers in this slice of arXiv.
Jian Ge
In 1986, Gromov asked whether every complete noncompact n-dimensional Riemannian manifold with nonnegative Ricci curvature and scalar curvature at least one satisfies: _g (B(p, R))≤ C_nRⁿ⁻² for all p∈M and R>0
Xinbao Lu, Kaiwen Yang
Banach asked in 1932 whether a real Banach space X whose n-dimensional subspaces, for some fixed 1<n<dimX, are all isometric must be a Hilbert space.Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov's even-dimensional result, this completes Banach's isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.
Boris Hasselblatt, JinCheng Wang
We study weakly hyperbolic magnetic (or twisted geodesic) flows of negatively curved surfaces (with nonpositive ''magnetic curvature") with a view to topological dynamics and ergodic theory, using their large-scale geometry on the universal cover.
Kotaro Motegi
We prove the existence of a genuinely time-dependent Brakke flow starting from Γ0⊂Rn+1 whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant L2 distance of Γ0 from an n-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of Γ0, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from Γ0. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.
Ari Krishna
We construct a smooth canonically polarized threefold, not biholomorphic to a product of positive-dimensional varieties, whose normalized Kähler-Einstein metric admits a non-integrable infinitesimal Einstein deformation. The same tangent direction is non-integrable as an infinitesimal complex deformation. In fact, the space of infinitesimal Einstein deformations in our example has real dimension 8, its integrable directions form a real 6-dimensional subspace, and every direction outside that subspace is obstructed. This answers both parts of a suitably generalized version of a question posed by Dai, Wang, and Wei in real dimension 6.
David Wiygul
For each integer k≥3 Hermann Karcher identified a complete singly periodic minimal surface Ξk (unique up to similarity) with 2k ends asymptotic to the union of k planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing Ξk,m for the quotient of Ξk by translation through m≥1 fundamental periods, we study the Morse index and nullity of the subfamilies Ξk,2 and Ξ3,m. In the m=2 case we prove for all k≥3 that Ξk,2 has Morse index 4k−3 and nullity 3. In the k=3 case we prove that there exists a real number α∗∈(1/3,1/2) such that for all m≥1 the Morse index of Ξ3,m is 6m−4⌊mα∗⌋−3 and its nullity is 3 unless mα∗ is an integer, in which case its nullity is 7.
Haohao Wang
A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen--Zheng, while the balanced fourfold case was settled recently by Wang--Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that a compact BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then g is Kähler.
Lauri Oksanen, Rakesh, Mikko Salo
We prove uniqueness in the Lorentzian Calderón problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted L2-estimates.
Hui Ma, Jiaxu Ma, Mingxuan Yang
We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect n∣h∣2−H2 with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.
Tymon Frelik, Wojciech Kryński
Cone structures on differentiable manifolds are fields of cones in the tangent bundle. A cone structure is isotrivial, modeled on a fixed immersed submanifold of the projective space, if at each point the projectivised cone is projectively equivalent to that submanifold. We consider cone structures arising from ordinary differential equations via a canonical construction. We prove that isotrivial cone structures in this class, modeled on generic curves in the n-dimensional projective space or ruled surfaces in the three-dimensional projective space, are flat. We also discuss applications to causal geometries in four dimensions and dispersionless Lax systems.
Ping Li
We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension n≥4 (assuming b2=1 when n≥6), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano n-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.
Fusheng Deng, Gaofeng Huang, Frank Kutzschebauch +1
We show that every diffeomorphism of Rn for n≥2 can be approximated by an automorphism of Cn in the Whitney Ck-topology for any positive integer k using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.
Rama Cont
We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds and define pathwise integrals for exact differential forms along such paths. The regularity of paths is defined in terms of p-th order variation along a sequence of partitions, for arbitrary integer p. We define the p-th variation tensor of a path along a sequence of partitions as a local symmetric tensor measure along the path, and derive a change of variable formula for smooth functions of paths with finite p-th variation. For p=2, our results extend H. Föllmer's pathwise Itô calculus to manifold-valued paths. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced p-jet of the test function, whose canonical highest-order component is determined by the p-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.
Benjamin Gess, Johannes Müller
A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.
Aleksandra Borówka, Zuzanna Szancer
For an almost contact pseudo-metric manifold we study a variation of the metric in the spirit of Meli-Ngakeu-Olea. Such variation is defined using a vector field on the manifold and the procedure shifts the signature of the metric by increasing the positive index by two. First we derive conditions on the vector field for the obtained manifold to remain almost contact pseudo-metric. Then we study the cases of almost pseudo co-Kähler manifolds. In particular we obtain geometric characterizations of variations for which the fundamental form remains closed and establish a local decomposition into a product. We further investigate the variation in the setting of statistical structures.
Mohammad Ghomi
We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.
Zuoqin Wang, Hanzhang Yun
Let Λ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee--Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin--Wodzicki residues of Λ and Λ2 explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.
Pingxin Gu, Haizhong Li, Yao Wan
In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space Hn for n≥3. We note that when n≥4, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.
Chaoqun Gao, Yong Wei, Rong Zhou
We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken--Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss--Green formula for divergence-measure fields.
Otis Chodosh, Matilde Gianocca
We characterize minimizers in Gamow's liquid drop problem.