arXiv research

Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.

In this paper, we generalize a theorem {à} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .

The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…

New topology defined from spacetime paths, reconstructing spacetime structure.

New method proves Alexandrov theorem for curved spacetimes.

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.

The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n >2, as shown by Malament and Hawking-King-McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spa…

In a previous paper we obtained formulae for the volume of a causal diamond or Alexandrov open set $I^+(p) \cap I^-(q)$ whose duration $τ(p,q)$ is short compared with the curvature scale. In the present paper we obtain asymptotic formulae valid when the point $q$ recedes to the future boundary ${\cal I}^+$ of an asymp…

Unified framework for Alexandrov 3-spaces, extending manifold results.

The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.

Alexandrov spaces have a special stratification that maps to spheres.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…

Generalizes Toponogov theorem to Alexandrov spaces.

Generalizes a soul-bound for noncompact Alexandrov spaces.

The paper improves collapsing Alexandrov spaces results using good coverings.

Graph comparison ties to Alexandrov's theorems.

Introduces Alexandrov spaces with curvature below, covering various theorems.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

Study quantifies convergence of Alexandrov spaces without collapsing.

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contra…

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

New definitions weaken conditions for Alexandrov spaces.

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.

Paper proves Toponogov's theorem in Alexandrov geometry.

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smoot…

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with re…

3D spaces without boundaries are found that aren't manifolds.

Study identifies topologies of 3D spaces with boundary.

We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.

This paper introduces Alexandrov's theory of singular surfaces and their curvature.

Sharp 3D Alexandrov inequality applied to volume-preserving flows.

Study open Alexandrov spaces with nonnegative curvature, proving structural results.

Proves Euler characteristic of collapsing Alexandrov spaces.

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.

Proves quantitative Alexandrov theorem for capillary surfaces.

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjectu…

Dans les années 1940-1970, Alexandrov et l'"École de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui est une mesure de Radon. Cette classe de surfaces a de bonnes prop…

We prove that any finite dimensional Alexandrov space with a lower curvature bound is locally Lipschitz contractible. As applications, we obtain a sufficient condition for solving the Plateau problem in an Alexandrov space considered by Mese and Zulkowski.

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting.

In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical…

We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.