Calculus of Variations and Geometric Measure Theory

T. Rajala

Local Poincaré inequalities from stable curvature conditions on metric spaces

created by rajala1 on 25 Oct 2011
modified on 14 Nov 2012


Published Paper

Inserted: 25 oct 2011
Last Updated: 14 nov 2012

Journal: Calc. Var. Partial Differential Equations
Volume: 44
Number: 3-4
Pages: 477-494
Year: 2012


We prove local Poincaré 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 Rényi entropy functional $\mathcal{E}_N(\rho m) = -\int_X \rho^{1-1/N} dm$ or the Shannon entropy functional $\mathcal{E}_\infty(\rho m) = \int_X \rho \log \rho dm$. We also prove that if the Rényi 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.

Tags: GeMeThNES
Keywords: Ricci curvature, Poincare inequality, Geodesics, metric measure spaces