On the geometry of metric measure spaces

TLDR

For Riemannian manifolds, lower Ricci curvature bounds coincide with classical Ricci curvature inequalities, and these bounds are stable under D‑convergence of metric measure spaces. The authors introduce and analyze lower (Ricci) curvature bounds for metric measure spaces. They define these bounds via convexity of relative entropy on the L2‑Wasserstein space and introduce a complete, separable length metric D on isomorphism classes of normalized metric measure spaces, interpretable through optimal mass transport. They prove that such curvature bounds imply volume‑growth estimates, establish the equivalence with Ricci curvature for manifolds, and show that families of spaces with bounded doubling constants (and diameter) are closed and compact under D‑convergence.

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.