Alexandrov spaces with maximal radius
type: Item
instance of: scholarly article
title: Alexandrov spaces with maximal radius
author: Karsten Grove
author: Peter Petersen
published in: Geometry \& Topology
publication date: 2022-12-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1805.10221
review text: The authors show several rigidity theorems for Alexandrov spaces with \(\operatorname{curv}\geq 1\), nonempty boundary and maximal radius (\(\operatorname{rad}=\frac{\pi}{2}\)). Assume that \(X\) is an \(n\)-dimensional Alexandrov space with \(\operatorname{curv}\geq 1\) and boundary \(\partial X \neq \emptyset\). Then there is a unique point \(s\) at maximal distance \(\leq \frac{\pi}{2}\) from \(\partial X\) called the soul point of \(X\). We denote by \(S_sX\) the space of directions at the soul \(s\) and by \(S^k(1)\) the \(k\)-dimensional unit sphere. More precisely, the authors show that: \begin{itemize} \item If \(\operatorname{rad}X=\frac{\pi}{2}\) and the soul of \(X\) is a regular point, then \(X\) is isometric to a spherical join \(S^k(1)\ast S\), where \(S\) is an \((n-k-1)\)-dimensional Alexandrov space with \(\operatorname{curv}\geq 1\), \(\partial S \neq \emptyset\) and \(\operatorname{rad}S < \frac{\pi}{2}\) (inner regularity); \item If \(\operatorname{rad}X =\frac{\pi}{2}\) and \(\operatorname{rad}S_sX>\frac{\pi}{2}\), then \(X=\hat{E}\ast\hat{S}\), where \(\hat{E}\) and \(\hat{S}\) are convex subsets of \(X\) with \(s\in \hat{S}\), \(\operatorname{rad}\hat{E} >\frac{\pi}{2}\), and \(\operatorname{rad}S_s\hat{S}>\frac{\pi}{2}\) (weak inner regularity). \end{itemize} The inner regularity theorem implies that if vol \(\partial X =\operatorname{vol}S^{(n-1)}(1)\), then \(X\) is isometric to an Alexandrov lens. The authors prove a topological regularity theorem under the assumption that \(\partial X\) is a topological manifold and a \(\mathbb{Z}_2\)-homology sphere. In the last part, the authors prove rigidity theorems for an Alexandrov space with \(\operatorname{curv}\geq k\) and \(\lambda_0\)-convex boundary (\(\lambda^2_0 > \max\{-k, 0\}\)).
reviewed by: Jialong Deng
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 7629608
zbMATH Keywords: Alexandrov geometry
zbMATH Keywords: rigidity
zbMATH Keywords: boundary convexity
zbMATH Keywords: maximal radius
MaRDI profile type: Publication
OpenAlex ID: W3003750512
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DOI: 10.2140/GT.2022.26.1635
description: scientific article; zbMATH DE number 7629608