Quantitative maximal volume entropy rigidity on Alexandrov spaces

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.