Lipschitz-Volume rigidity in Alexandrov geometry

Abstract

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f:X=⨿X_{\ell }\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.

This is the published version on AIM

Cited by in corpus (2)

• Sagitta, Lenses, and Maximal Volume
• Lipschitz-Volume rigidity on limit spaces with Ricci curvature bounded from below