Acta Mathematica

, Volume 196, Issue 1, pp 65–131

On the geometry of metric measure spaces

Authors

    • Institut für Angewandte MathematikUniversität Bonn
Article

DOI: 10.1007/s11511-006-0002-8

Cite this article as:
Sturm, K. Acta Math (2006) 196: 65. doi:10.1007/s11511-006-0002-8

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 L2-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.

Copyright information

© Springer-Verlag 2006