Ricci curvature for metric-measure spaces via optimal transport

John Lott; Cedric Villani

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Ricci curvature for metric-measure spaces via optimal transport

John Lott·Cedric Villani

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Abstract

We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N finite, or having infinity-Ricci curvature bounded below by K, for K a real number. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein 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.

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John Lott, Cedric Villani. 2006-06-23. Ricci curvature for metric-measure spaces via optimal transport. https://arxiv.org/abs/math/0412127

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