Quantitative maximal volume entropy rigidity on Alexandrov spaces

TL;DR

This paper proves a quantitative volume entropy rigidity result for Alexandrov spaces with curvature bounds, extending previous work to a broader class of metric spaces and also addressing RCD*-spaces in the non-collapsing case.

Contribution

It extends the quantitative maximal volume entropy rigidity theorem from Riemannian manifolds to Alexandrov spaces and RCD*-spaces, broadening the scope of geometric rigidity results.

Findings

Alexandrov spaces with curvature ≥ -1 and high volume entropy are close to hyperbolic manifolds.

The result applies to non-collapsing RCD*-spaces, showing rigidity in this setting.

Provides explicit epsilon bounds depending on dimension and diameter.

Abstract

We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $\epsilon(N, D)>0$, such that for $\epsilon<\epsilon(N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $\geq -1$, $\operatorname{diam}(X)\leq D, h(X)\geq N-1-\epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity of \cite{CRX} to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for $\op{RCD}^*$-spaces in the non-collapsing case.