Solves Alexandrov's problem for hyperbolic convex bodies.
problem Finding a convex body with a given curvature measure in hyperbolic space.
method Defined Gauss curvature measure, proved existence and uniqueness of solution.
result Uniqueness of the solution to Alexandrov's problem in hyperbolic space.
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.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.
Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by c…
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
Method solves optimisation problems on non-Riemannian surfaces with bilateral curvature bounds.
problem Optimisation problems on non-Riemannian surfaces with sharp edges.
method Forward-backward splitting in Alexandrov spaces with bilateral curvature bounds.
result Convergence of the forward-backward method in Alexandrov spaces with bilateral curvature bounds.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
Unified framework for Alexandrov 3-spaces, extending manifold results.
problem Extend manifold topology results to Alexandrov 3-spaces.
method Generalize connected sum, prime decomposition, and Dehn surgery.
result Unified framework for Alexandrov 3-spaces, including non-manifold spaces.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
problem Conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
method Proposes a Gluing Conjecture and proves it under certain conditions.
result The Gluing Conjecture is true under specific conditions, generalizing Petrunin's Gluing Theorem.
Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
The paper extends an Alexandrov theorem to Minkowski spacetime.
problem Generalizing Alexandrov's theorem to Minkowski spacetime.
method Adapting conditions for closed codimension-two spacelike submanifolds in Minkowski spacetime.
result A generalized Alexandrov theorem for Minkowski spacetime.
Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
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…
Classifies 4D Alexandrov spaces with torus actions.
problem Classifying Alexandrov spaces with torus actions.
method Equivariant classification and homeomorphism analysis.
result Alexandrov spaces are homeomorphic to Riemannian orbifolds.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
The paper improves collapsing Alexandrov spaces results using good coverings.
problem Collapsing Alexandrov spaces with no proper extremal subsets.
method Application of good coverings of Alexandrov spaces.
result Construction of an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups.
Graph comparison ties to Alexandrov's theorems.
problem Graph comparison conditions on metric spaces.
method Proof of Alexandrov's implications from graph comparisons.
result Complete description of graphs with trivial comparisons.
Introduces Alexandrov spaces with curvature below, covering various theorems.
problem Understanding spaces with curvature constraints.
method Explains comparison conditions, globalization, tangent spaces, etc.
result Globalization theorem and other theorems established for Alexandrov spaces.
The paper solves a curvature problem in hyperbolic space using a flow approach.
problem Prescribed Gaussian curvature problem in hyperbolic space.
method Flow approach to prove existence and uniqueness of solutions.
result Existence and uniqueness of solutions for α≥n+1. Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniforml…
Alexandrov immersed surfaces maintain their properties under mean curvature flow.
problem Preserving Alexandrov immersivity in mean curvature flow.
method Mean curvature flow techniques adapted for Alexandrov immersed, 2D surfaces.
result Mean curvature flow properties hold for Alexandrov immersed surfaces.
Study quantifies convergence of Alexandrov spaces without collapsing.
problem Quantifying convergence of Alexandrov spaces without collapsing.
method Lipschitz homotopy convergence for Alexandrov spaces.
result Lipschitz homotopies can be chosen to preserve singular strata.
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 Tn−1 action. In contra…
Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.
problem Quantifying rigidity in Alexandrov spaces with curvature constraints.
method Using Gromov-Hausdorff distance and properties of Alexandrov spaces.
result Alexandrov spaces with curvature bounds are close to hyperbolic manifolds.
New definitions weaken conditions for Alexandrov spaces.
problem Weak conditions for Alexandrov spaces with curvature bounds.
method Introducing imaginary comparison and angles, and right/left bounded second derivative.
result New proofs for Doubling and Globalization Theorems.
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.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
Paper proves Toponogov's theorem in Alexandrov geometry.
problem Proving Toponogov's theorem in Alexandrov geometry with lower curvature bound.
method Inspired by Riemannian geometry, uses second variation formula.
result Elementary proof of 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:X=⨿Xℓ→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…
In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become tot…
3D spaces without boundaries are found that aren't manifolds.
problem Finding compact aspherical Alexandrov spaces without boundaries.
method Constructing specific examples of 3D spaces.
result Examples of 3D spaces without boundaries that are not topological manifolds.
Study identifies topologies of 3D spaces with boundary.
problem Understanding the topologies of compact Alexandrov spaces with boundary.
method Continuation of previous work, determining topologies through analysis.
result Identified topologies of collapsing 3D Alexandrov 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.
problem Understanding the geometry of singular surfaces with intrinsic metrics and curvature.
method Develops the theory of Alexandrov surfaces, focusing on their convergence and stability properties.
result Classifies compact Alexandrov surfaces using the conformal viewpoint introduced by Reshetnyak.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
Study open Alexandrov spaces with nonnegative curvature, proving structural results.
problem Understanding open Alexandrov spaces with nonnegative curvature.
method Establishing structural results on open Alexandrov spaces.
result Structural results on open Alexandrov spaces with nonnegative curvature.
Proves Lipschitz regularity of harmonic maps from Alexandrov spaces.
problem Lipschitz regularity of harmonic maps from finite dimensional Alexandrov spaces.
method Extends Huang-Wang's argument to finite dimensional Alexandrov spaces.
result Solves F. H. Lin's conjecture on harmonic maps.
Defines Alexandrov spaces via axioms, focusing on curvature bounds.
problem Understanding Alexandrov spaces with curvature constraints.
method Formalizes spaces via axioms, distinguishing between curvature above and below.
result Comprehensive structure theory of Alexandrov spaces with curvature bounds.
Proves Euler characteristic of collapsing Alexandrov spaces.
problem Euler characteristic of collapsing Alexandrov spaces.
method Analyzes strata and fibers of the limit space.
result Euler characteristic equals sum of products of strata and fiber Euler characteristics.
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.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
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…