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arXiv · 1911.00120

Lipschitz-Volume Rigidity and Globalization

Nan Li
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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.

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Nan Li. 2019-10-31. Lipschitz-Volume Rigidity and Globalization. https://arxiv.org/abs/1911.00120

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