Bacher, Kathrin: Curvature-Dimension Bounds and Functional Inequalities : Localization, Tensorization and Stability. - Bonn, 2010. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
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author = {{Kathrin Bacher}},
title = {Curvature-Dimension Bounds and Functional Inequalities : Localization, Tensorization and Stability},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2010,
month = mar,

note = {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.
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.
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.
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.
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).},

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