Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 114, 14 pages      arXiv:2006.04092      https://doi.org/10.3842/SIGMA.2020.114
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

The Measure Preserving Isometry Groups of Metric Measure Spaces

Yifan Guo ab
a)  Beijing Institute of Mathematical Sciences and Applications, Beijing, P.R. China
b)  Department of Mathematics, University of California, Irvine, CA, USA

Received June 30, 2020, in final form November 02, 2020; Published online November 10, 2020

Abstract
Bochner's theorem says that if $M$ is a compact Riemannian manifold with negative Ricci curvature, then the isometry group $\operatorname{Iso}(M)$ is finite. In this article, we show that if $(X,d,m)$ is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group $\operatorname{Iso}(X,d,m)$ is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Émery Ricci curvature except for small portions.

Key words: optimal transport; synthetic Ricci curvature; metric measure space; Bochner's theorem; measure preserving isometry.

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