Local Poincar\'e inequalities from stable curvature conditions on metric spaces

TL;DR

This paper establishes local Poincaré inequalities in metric spaces under stable curvature conditions, including weak CD(K,N) spaces and those with specific entropy flow properties, advancing understanding of geometric analysis in these contexts.

Contribution

It proves local Poincaré inequalities under stable curvature-dimension conditions and explores implications for geodesic uniqueness in metric measure spaces.

Findings

Poincaré inequalities hold under weak CD(K,N) conditions.

Flow conditions imply geodesic uniqueness at almost all points.

Stability under measured Gromov-Hausdorff convergence is demonstrated.

Abstract

We prove local Poincar\'e inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the R\'enyi entropy functional $E_N(\rho m) = -\int_X \rho^{1-1/N} dm$ or the Shannon entropy functional $E_\infty(\rho m) = \int_X \rho \log \rho dm$. We also prove that if the R\'enyi entropy functional is strongly displacement convex in the Wasserstein space, then at every point of the space we have unique geodesics to almost all points of the space.