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Alexandrov spaces with maximal radius

Karsten Grove and Peter Petersen

Geometry & Topology 26 (2022) 1635–1668
Abstract

We prove several rigidity theorems related to and including Lytchak’s problem. The focus is on Alexandrov spaces with curv 1, nonempty boundary and maximal radius π 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, 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.

Keywords
Alexandrov geometry, rigidity, boundary convexity
Mathematical Subject Classification 2010
Primary: 53C20, 53C24
Secondary: 53C23
References
Publication
Received: 4 March 2020
Revised: 5 April 2021
Accepted: 7 May 2021
Published: 28 October 2022
Proposed: Paul Seidel
Seconded: John Lott, Dmitri Burago
Authors
Karsten Grove
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
Peter Petersen
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States