Lipschitz-Volume Rigidity and Globalization

TL;DR

This paper surveys Lipschitz-Volume Rigidity theorems in singular spaces with curvature bounds, highlighting when 1-Lipschitz maps preserve path length and discussing open problems in the field.

Contribution

It provides a comprehensive overview of Lipschitz-Volume Rigidity in singular spaces with curvature bounds and explores related open questions.

Findings

Lipschitz-Volume Rigidity holds for certain singular spaces with curvature bounds.

Conditions under which 1-Lipschitz maps preserve path length are characterized.

Open problems remain in extending rigidity results to broader classes of singular spaces.

Abstract

Let $X$ and $Y$ be length metric spaces. Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\colon X\to Y$ and $0<\mathcal H^n(X)=\mathcal H^n(f(X))<\infty$, then $f$ preserves the length of path. This property holds for smooth manifolds but doesn't hold for all singular spaces. We survey the Lipschitz-Volume Rigidity Theorems on singular spaces with lower curvature bounds and discuss some related open problems.