The Measure Preserving Isometry Groups of Metric Measure Spaces
Yifan Guo

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 is a compact Riemannian manifold with negative Ricci curvature, then the isometry group is finite. In this article, we show that if is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group 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.
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