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

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.

63 pages. We have revised the article to correct omissions and mistakes in the original version