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 N, D , there exists ϵ ( N , D ) > 0 \epsilon (N, D)>0 , such that for ϵ > ϵ ( N , D ) \epsilon >\epsilon (N, D) , if X X is an N N -dimensional Alexandrov space with curvature ≥ − 1 \geq -1 , diam ⁡ ( X ) ≤ D , h ( X ) ≥ N − 1 − ϵ \operatorname {diam}(X)\leq D, h(X)\geq N-1-\epsilon , then X X is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity provided by Chen, Rong, and Xu [J. Differential Geom. 113 (2019), pp. 227–272] to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for RCD ∗ \operatorname {RCD}^* -spaces in the non-collapsing case.