Ricci curvature for metric-measure spaces via optimal transport

We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N [1, ), or having -Ricci curvature bounded below by K, for K 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. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature…

• NS
  National Science Foundation

  Award: DMS-0306242
• CM
  Clay Mathematics Institute