arXiv research

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

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

Introduces Alexandrov spaces with curvature below, covering various theorems.

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.

Paper proves Toponogov's theorem in Alexandrov geometry.

Generalizes a soul-bound for noncompact Alexandrov spaces.

New definitions weaken conditions for Alexandrov spaces.

Paper proves curvature conditions are preserved in metric spaces.

Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.

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…

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.

Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.

Proves curvature tensor convergence for smoothable spaces.

Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.

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

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…

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.

New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.

Here I show compatibility of two definition of generalized curvature bounds --- the lower bound for sectional curvature in the sense of Alexandrov and lower bound for Ricci curvature in the sense of Lott--Villani--Sturm.

Paper proves Allard's theorem in Alexandrov spaces.

In this paper we give a new proof for an almost isometry theorem in Alexandrov spaces with curvature bounded below.

Characterizes orbifolds with upper curvature bounds as reflectofolds.

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

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

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.

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

We show that on every ${\sf RCD}$ spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are ${\sf RCD}$ spaces, our construction applies in particular to the Alexandrov setting.…

Study flow on de Sitter space for convex hypersurfaces.

We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-ma…

We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

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.

Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces ar…

Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ro…

We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov.

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of t…

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.

Negative curvature proven in Sasaki manifold space completion.

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.

We prove the existence of two Alexandrov embedded closed magnetic geodesics on any two dimensional sphere with nonnegative Gauss curvature.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

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

Paper shows equivalence of two curvature notions on singular surfaces.

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

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

Survey on gluing constructions under lower curvature bounds.