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Volume entropy and rigidity for RCD-spaces

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

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.

Volume entropy and rigidity for RCD-spaces

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 from a non-collapsed RCD space without boundary to a locally symmetric -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 is a locally symmetric manifold, and is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of and involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on -orbifolds.

Paper Structure

This paper contains 22 sections, 39 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The barycenter technique and organization of the paper
  3. Preliminaries
  4. $\mathop{\mathrm{\mathsf{RCD}}}\nolimits$ spaces and their boundary
  5. The Coarea formula and the definition of the Jacobian matrix
  6. Structure of spaces that are both
  7. Sobolev to Lipschitz Property for Maps
  8. Bounds on the index invariant of maps from spaces
  9. The average number of preimages
  10. Bounds between the
  11. Properties of the barycenter and natural maps
  12. Barycenters
  13. The natural maps
  14. Proof of inequality cases in Theorems \ref{['thm:entrig']} and \ref{['thm:schwarz']}
  15. Rigidity Cases
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $K \in {\bf R}$, $N \in {\bf N}$ with $N\geq 3$, $(X,d,\mathcal{H}^{N})$ be an $\mathop{\mathrm{\mathsf{RCD}}}\nolimits(K,N)$ space without boundary, and $Y$ be a closed orientable negatively curved locally symmetric space of dimension $N$. Then, for any continuous map $f:X\to Y$, Moreover, if we have equality and $\mathop{\mathrm{ind}}\nolimits_H(f)=\mathop{\mathrm{ind}}\nolimits_\pi(f)$, th

Theorems & Definitions (85)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: Lipschitz volume Rigidity
  • Definition 2.1
  • Theorem 2.2: MNDePhilippisGigliGigliMondinoRajalaKapovitch-MondinoAmbrosioHondaTewodrosebrue-semola:18
  • ...and 75 more