Curvature-Dimension Bounds and Functional Inequalities

Abstract

This work is devoted to the analysis of abstract metric measure spaces (M,d,m) satisfying the curvature-dimension condition CD(K,N) presented by Sturm and in a similar form by Lott and Villani.<br /> In the first part, we introduce the notion of a Borell-Brascamp-Lieb inequality in the setting of metric measure spaces denoted by BBL(K,N). This inequality holds true on metric measure spaces fulfilling the curvature-dimension condition CD(K,N) and is stable under convergence of metric measure spaces with respect to the transportation distance. <br /> In the second part, 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 one. This so-called reduced curvature-dimension condition CD*(K,N) has the localization property. Furthermore, we show its stability and the tensorization property.<br /> 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. <br /> In the third part, we study cones over metric measure spaces. We deduce that the n-Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the n-spherical cone over the same manifold fulfills CD(n,n+1).