The Measure Preserving Isometry Groups of Metric Measure Spaces

TL;DR

This paper extends classical results on isometry groups from Riemannian manifolds to metric measure spaces with synthetic negative Ricci curvature, showing finiteness of measure-preserving isometries and providing estimates.

Contribution

It proves the finiteness of measure-preserving isometry groups for metric measure spaces with synthetic negative Ricci curvature and offers effective bounds for weighted Riemannian manifolds.

Findings

Measure-preserving isometry group is finite for spaces with synthetic negative Ricci curvature.

Provides an effective estimate on the order of the isometry group for weighted Riemannian manifolds.

Extends classical Riemannian results to more general metric measure spaces.

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-\'Emery Ricci curvature except for small portions.