Alexandrov spaces with maximal radius
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 (\(\lambda^2_0 > \max\{-k, 0\}\)).
scientific article; zbMATH DE number 7629608
scientific article