Lipschitz-Volume Rigidity and Globalization

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 $f:X\to Y$ 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.