, Volume 196, Issue 1, pp 65-131

On the geometry of metric measure spaces

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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.