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2011

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe

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In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries.

It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow.
Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties.
In these spaces, that we call RCD(K,\infty) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L^\infty-Lip Feller regularization.

Metric measure spaces with Riemannian Ricci curvature bounded from below · Around

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