## Lipschitz-Volume Rigidity and Globalization

**Speaker(s): ** Nan Li (CUNY)

**Time: ** 09:00-10:00 July 6, 2019

**Venue: ** Room 29, Quan Zhai, BICMR

Abstract: Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. Given $n$-dimensional length metric spaces $X$ and $Y$, the Lipschitz-Volume Rigidity stands for the 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 (singular) spaces with lower curvature bounds, but doesn't hold in general.

In this talk, globalization means to prove a space to be an Alexandrov space assuming local curvature comparisons on an open dense subset. This can't be done for free since a concave open domain in $\dR^n$ is always locally flat, but its closure is not an Alexandrov space.

We will discuss the results regarding the above two problems and the connections between them. We will also report the related projects that we are working on.