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On the geometry of metric measure spaces

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Acta Mathematica

Abstract

We introduce and analyze lower (Ricci) curvature bounds \( \underline{{Curv}} {\left( {M,d,m} \right)} \) ⩾ K for metric measure spaces \( {\left( {M,d,m} \right)} \). Our definition is based on convexity properties of the relative entropy \( Ent{\left( { \cdot \left| m \right.} \right)} \) regarded as a function on the L 2-Wasserstein space of probability measures on the metric space \( {\left( {M,d} \right)} \). Among others, we show that \( \underline{{Curv}} {\left( {M,d,m} \right)} \) ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, \( \underline{{Curv}} {\left( {M,d,m} \right)} \) ⩾ K if and only if \( Ric_{M} {\left( {\xi ,\xi } \right)} \) ⩾ K \( {\left| \xi \right|}^{2} \) for all \( \xi \in TM \).

The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.

We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.

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  1. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, DE-53115, Bonn, Germany

    Karl-Theodor Sturm

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Sturm, KT. On the geometry of metric measure spaces. Acta Math 196, 65–131 (2006). https://doi.org/10.1007/s11511-006-0002-8

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