# Ricci curvature for metric-measure spaces via optimal transport

@article{Lott2004RicciCF,
title={Ricci curvature for metric-measure spaces via optimal transport},
author={J. Lott and C. Villani},
journal={Annals of Mathematics},
year={2004},
volume={169},
pages={903-991}
}
• Published 2004
• Mathematics
• Annals of Mathematics
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about… Expand
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#### References

SHOWING 1-10 OF 63 REFERENCES
Lower bounds on Ricci curvature and the almost rigidity of warped products
• Mathematics
• 1996
The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem,Expand
On the structure of spaces with Ricci curvature bounded below. II
• Mathematics
• 2000
In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a deExpand
Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
• Mathematics
• 2006
We investigate Prekopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e V where the potential V and the Ricci curvature satisfy Hessx V + Ricx I for all xExpand
Gradient flows with metric and differentiable structures, and applications to the Wasserstein space
• Mathematics
• 2004
— In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following someExpand
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
• Mathematics
• 2001
Abstract.A concavity estimate is derived for interpolations between L1(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell,Expand
QUASIGEODESICS AND GRADIENT CURVES IN ALEXANDROV SPACES
• 2003
1. A comparison theorem for complete Riemannian manifolds with sectional curvatures ≥ k says that distance functions in such manifolds are more concave than in the model space Sk of constantExpand
Transport inequalities, gradient estimates, entropy and Ricci curvature
• Mathematics
• 2005
We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on theExpand
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
• Mathematics
• 2005
Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the ConvergenceExpand
On the geometry of metric measure spaces. II
AbstractWe introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound \$\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m}Expand
Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
The three-dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations but can also be seen, following Arnold [1], as a geodesic on a group ofExpand