Optimal Transport and Ricci Curvature for Metric- Measure Spaces

Abstract

We survey work of Lott-Villani and Sturm on lower curvature bounds for metric-measure spaces. An intriguing question is whether one can extend notions of smooth Riemannian geome- try to general metric spaces. Besides the inherent interest, such extensions sometimes allow one to prove results about smooth Riemannian manifolds, using compactness theorems. There is a good notion of a metric space having sectional curvature bounded below by or sectional curvature bounded above by K, due to Alexandrov. We refer to the articles of Petrunin and Buyalo-Schroeder in this volume for further information on these two topics. In this article we address the issue of whether there is a good notion of a metric space having Ricci curvature bounded below by A motivation for this question comes from Gromov's precompactness theorem (14, The- orem 5.3). Let M denote the set of compact metric spaces (modulo isometry) with the Gromov-Hausdorff topology. The precompactness theorem says that given N ∈ Z + , D < ∞ and K ∈ R, the subset of M consisting of closed Riemannian manifolds (M, g) with dim(M) = N, Ric ≥ Kg and diam ≤ D, is precompact. The limit points in M of this subset will be metric spaces of Hausdorff dimension at most N, but generally are not manifolds. However, one would like to say that in some generalized sense they do have curvature bounded below by K. Deep results about the structure of such limit points, which we call limits, were obtained by Cheeger and Colding (8, 9, 10). We refer to the article of Guofang Wei in this volume for further information. In the work of Cheeger and Colding, and in earlier work of Fukaya (13), it turned out to be useful to consider not just metric spaces, but rather metric spaces equipped with measures. Given a compact metric space (X, d), let P(X) denote the set of Borel probability measures on X. That is, ν ∈ P(X) means that ν is a nonnegative Borel measure on X with R X dν = 1. We put the weak-∗ topology on P(X), so limi→∞ νi = ν if and only if for all f ∈ C(X), we have limi→∞ R X f dνi = R X f dν. Then P(X) is compact. Definition 0.1. A compact metric measure space is a triple (X, d, ν) where (X, d) is a compact metric space and ν ∈ P(X). Definition 0.2. Given two compact metric spaces (X1, d1) and (X2, d2), an ǫ-Gromov- Hausdorff approximation from X1 to X2 is a (not necessarily continuous) map f : X1 → X2 so that