# Maximal Volume Entropy Rigidity for $RCD^*(-(N-1),N)$

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Raquel Perales

created by daniele on 23 Oct 2018

modified on 06 Dec 2018

6 dec 2018
-- 15:00

Aula Tricerri, DiMaI, Firenze

**Abstract.**

Maximal Volume Entropy Rigidity for $RCD^*(-(N-1),N)$
(with Connell, Dai, Nunez-Zimbron, Suarez-Serrato, Wei)

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 show the same kind of maximal entropy rigidity holds for a class of metric measure spaces known by now as *$RCD^*(K,N)$ spaces*. While the upper bound follows quickly, the rigidity case is quite involved due to the lack of a smooth structure in $RCD^*$ spaces.