Volume entropy and rigidity for RCD-spaces

Chris Connell; Xianzhe Dai; Jesús Núñez-Zimbrón; Raquel Perales; Pablo Suárez-Serrato; Guofang Wei

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arXiv · 2411.04327

Volume entropy and rigidity for RCD-spaces

Chris Connell·Xianzhe Dai·Jesús Núñez-Zimbrón·Raquel Perales·Pablo Suárez-Serrato·Guofang Wei

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Abstract

We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds.

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Chris Connell, Xianzhe Dai, Jesús Núñez-Zimbrón, Raquel Perales, Pablo Suárez-Serrato, Guofang Wei. 2024-11-07. Volume entropy and rigidity for RCD-spaces. https://arxiv.org/abs/2411.04327

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