Ricci curvature for metric-measure spaces via optimal transport

Abstract

We define a notion of a measured length space $X$ having nonnegative $N$-Ricci curvature, for $N \in [1, \infty)$, or having $\infty$-Ricci curvature bounded below by $K$, for $K \in \mathbb{R}$. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space $P_2(X)$ of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.

Authors

John Lott

Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
United States

Cedric Villani

UMPA (UMR CNRS 5669)
École Normale Supérieure de Lyon
69364 Lyon
France