Ricci curvature for metric-measure spaces via optimal transport

2009/04/22 by John Lott, Cédric Villani, Cedric Villani · 1,298 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Computer science #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Measure (data warehouse) #Metric (unit) #Metric space #Pure mathematics #Ricci curvature #Ricci flow #Scalar curvature #Topology (electrical circuits)

paper · pdf · doi:10.4007/annals.2009.169.903

published in Annals of Mathematics 169(3), 903-991 (Princeton University)

openalex publication_date 2009/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

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 bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

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