Alexandrov spaces with maximal radius

Grove, Karsten¹ ; Petersen, Peter²

¹ Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States
² Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, United States

Geometry & topology, Tome 26 (2022) no. 4, pp. 1635-1668.

Résumé

We prove several rigidity theorems related to and including Lytchak’s problem. The focus is on Alexandrov spaces with $\mathrm{curv}\ge 1$, 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, 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.

Détail

DOI : 10.2140/gt.2022.26.1635

Classification : 53C20, 53C24, 53C23
Keywords: Alexandrov geometry, rigidity, boundary convexity

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