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Acta Mathematica

, Volume 196, Issue 1, pp 65–131 | Cite as

On the geometry of metric measure spaces

  • Karl-Theodor Sturm
Article

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.

Keywords

Measure Space Heat Kernel Relative Entropy Ricci Curvature Dirichlet Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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