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.