Alexandrov Spaces with Maximal Radius

TL;DR

This paper investigates Alexandrov spaces with curvature ≥1 and maximal radius, establishing rigidity theorems under certain boundary conditions and highlighting the class's flexibility and geometric constraints.

Contribution

It proves new rigidity theorems for Alexandrov spaces with maximal radius, especially when boundaries are spherical, and explores the space's flexibility and curvature constraints.

Findings

Spaces with spherical boundary exhibit strong rigidity.

General lower curvature bounds restrict maximal radius to cones.

Connections to positive mass conjectures are discussed.

Abstract

Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \frac{\pi}{2}. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that when the boundary is either geometrically or topologically spherical, then it is possible to obtain strong rigidity results. In contrast to this one can show that with general lower curvature bounds and strictly convex boundary only cones can have maximal radius. We also mention some connections between our problems and the positive mass conjectures. This paper is an expanded version and replacement of the two previous versions