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.
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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…
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.
Estimates volume of convex Alexandrov spaces with boundary.
problem Estimating volume of Alexandrov spaces with convex boundaries.
method Gradient flow of semi-concave functions.
result Volume upper bound achieved implies Boundary Conjecture.
New integral estimates on substatic manifolds improve Alexandrov Theorem.
problem Improving integral estimates on substatic manifolds.
method Introducing a new vector field with nonnegative divergence.
result Generalization and improvement of integral estimates leading to Alexandrov Theorem.
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.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
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…
Estimates complex Hessian integral for complex Monge-Ampère equations.
problem Improving classical ABP estimate for complex settings.
method De Giorgi iteration method for complex Monge-Ampère equations.
result Sharp gradient estimates for complex Monge-Ampère equations.
We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.
New proof of Kähler-Einstein Fano manifold estimates.
problem Uniform estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds.
method Using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle, without pluripotential theory.
result Uniform estimates derived for Kähler-Ricci flow on Kähler-Einstein Fano manifolds.
The paper measures and limits the extent of non-smooth points in Alexandrov spaces.
problem Understanding the extent of non-smooth points in Alexandrov spaces.
method Defining a non-negative function to measure the extent of non-smoothness and quantitatively estimating its distribution.
result The Hausdorff dimension estimate and quantitative Hausdorff measure estimate for the set of -singular points.
Paper doubles Hessian estimates for special Lagrangian equation with constraints.
problem Estimating Hessian for special Lagrangian equation under general phase constraints.
method Doubling argument, Alexandrov-type theorems.
result Established Hessian estimates for special Lagrangian equation.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
In this note, we study the radius of positively curved or non-negatively curved Alexandrov space with strictly convex boundary, with convexity measured by the Base-Angle defined by Alexander and Bishop. We also estimate the volume of the boundary of non-negatively curved spaces as well as the rigidity case, which can b…
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
For a path in a compact finite dimensional Alexandrov space with curv , the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of , the dimension, di…
This is the first paper of two ones. Here we prove that two compact Alexandrov surfaces of bounded integral curvature having no peak points are bi-Lipschitz equivalent if they are homeomorphic one to the other. Also conditions under that two ends having finite integral negative curvature are bi-Lipschitz equivalent are…
Sharp stability of Alexandrov's theorem for domains in the small-excess regime
problem Stability of Alexandrov's theorem for domains in the small-excess regime
method Combines a version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region
result Sharp stability estimate in a genuinely non-parametric regime
Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalizati…
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 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.
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…
In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the -dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for we obtain a Minkow…
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 estimates singular sets in Alexandrov spaces and proves packing and Hausdorff measure estimates.
problem Estimating singular sets in Alexandrov spaces with curvature bounded below.
method Analyzing -scale -singular sets and using packing estimates to derive Hausdorff measure bounds.
result Hausdorff measure estimates for singular sets in Alexandrov spaces.
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, -dimensional, cohomogeneity one Alexandrov spaces admitting an isometric action. In contra…
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.
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.
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
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.
Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.
problem Studying spaces with non-Riemannian curvature beyond Alexandrov geometry.
method Introducing a weak quadruple comparison principle and developing a strainer theory.
result Spaces have constant integer dimension, measure contraction property, and unique Banach tangent cones.
The paper improves stability estimates for soap bubble theorem in curved domains.
problem Stability estimates for the Soap Bubble Theorem in curved domains.
method Leveraging Gagliardo-Nirenberg-type interpolation inequalities.
result Optimal stability estimates for deviations of mean curvature from being constant.
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.
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.
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of . We furthermore characterize the metric structure on with re…
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 -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 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.