Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces

TL;DR

This paper investigates the properties of the curvature-dimension condition in metric measure spaces, establishing local-to-global equivalence, tensorization, and implications for the fundamental group under positive curvature.

Contribution

It proves the equivalence of local and global curvature-dimension conditions and demonstrates tensorization for the reduced condition CD*(K,N).

Findings

Local CD(K,N) is equivalent to global CD*(K,N)

CD*(K,N) has the local-to-global property

Fundamental group is finite under positive curvature conditions

Abstract

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD*(K,N) has the local-to-global property. We also prove the tensorization property for CD*(K,N). As an application we conclude that the fundamental group of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N.