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Vitali Kapovitch

Publications and source records attributed to Vitali Kapovitch.

At least 19 recordsLinked to original sources
arXiv2026

Lectures on Alexandrov spaces with curvature bounded below

An introduction to Alexandrov spaces with curvature bounded below. Topics include various comparison conditions, the globalization theorem, tangent spaces and spaces of directions, gradient flows, the splitting theorem, dimension and volume, Gromov's selection theorem, the boundary and the doubling theorem, and quotient spaces. We also give a brief overview of the two-dimensional theory, the main precursor to modern Alexandrov geometry.

math.DG↗
arXiv2026

An invitation to Alexandrov geometry: CAT(0) spaces

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 globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.

math.DG↗
arXiv2025

Alexandrov meets Kirszbraun

We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder. We also discuss related questions, both solved and open.

math.DG↗
arXiv2023

Alexandrov geometry: foundations

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 are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering the two contexts simultaneously. We only consider the intrinsic theory, leaving applications aside. This book includes material up to the definition of dimension. Another volume still in preparation will cover further topics.

math.DG↗
arXiv2022

On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds

We show that if an Alexandrov space $X$ has an Alexandrov subspace $\bar Ω$ of the same dimension disjoint from the boundary of $X$, then the topological boundary of $\bar Ω$ coincides with its Alexandrov boundary. Similarly, if a noncollapsed RCD(K,N) space $X$ has a noncollapsed RCD(K,N) subspace $\bar Ω$ disjoint from boundary of $X$ and with mild boundary condition, then the topological boundary of $\bar Ω$ coincides with its De Philippis-Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.

math.MG↗
arXiv2020

Metric-measure boundary and geodesic flow on Alexandrov spaces

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

math.DG↗
arXiv2020

Structure of Submetries

We investigate the geometric and topological structure of equidistant decompositions of Riemannian manifolds.

math.DG↗
arXiv2020

On the topology and the boundary of N-dimensional RCD(K,N) spaces

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.

math.MG↗
arXiv2020

On gluing Alexandrov spaces with lower Ricci curvature bounds

In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling and gluing constructions.

math.DG↗
arXiv2019

On the structure of RCD spaces with upper curvature bounds

We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are DC coordinates and the distance is induced by a continuous BV Riemannian metric.

math.DG↗
arXiv2019

Nilpotency, almost nonnegative curvature and the gradient flow

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

math.DG↗
arXiv2018

CD meets CAT

We show that if a noncollapsed $CD(K,n)$ space $X$ with $n\ge 2$ has curvature bounded above by $κ$ in the sense of Alexandrov then $K\le (n-1)κ$ and $X$ is an Alexandrov space of curvature bounded below by $K-κ(n-2)$. We also show that if a $CD(K,n)$ space $Y$ with finite $n$ has curvature bounded above then it is infinitesimally Hilbertian.

math.DG↗
arXiv2017

On noncollapsed almost Ricci-flat 4-manifolds

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.

math.DG↗
arXiv2017

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over the product of C and T admit no complete nonnegatively curved metric.

math.DG↗