Maximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spaces

TL;DR

This paper extends the maximal volume entropy rigidity result from Riemannian manifolds to $ ext{RCD}^*$ spaces, showing that equality characterizes hyperbolic spaces even in non-smooth settings.

Contribution

It generalizes the volume entropy rigidity theorem to $ ext{RCD}^*$ spaces, overcoming challenges due to the lack of smooth structure.

Findings

Established volume entropy upper bound for $ ext{RCD}^*$ spaces.

Proved rigidity characterization of hyperbolic spaces in this setting.

Derived an almost rigidity result similar to Cheng-Rong-Xu's for Riemannian manifolds.

Abstract

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces. As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds.