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... of the Boltzmann entropy (or exponential contractivity of the heat semigroup with respect to 2-Wasserstein distance) to lower bounds for the Ricci curvature on a Riemannian manifold. Sturm [46] [47] =-=[48]-=- and Lott & Villani [38] extended this idea to nonlinear evolution equations, and to a means of defining Ricci curvature bounds in abstract metric-measure spaces. 4 JOSÉ A. CARRILLO, ROBERT J. MCCANN...

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... exposed in our work are new. From the Riemannian point of view, our approach boils down to contraction of the Lipschitz norm by the heat equation, which is one of the results of Bakry and Émery ([5,6], see also [1] and [43]). This latter property was suggested in [43] as a possible definition of a lower bound on Ricci curvature for diffusion operators in general spaces, though it does not provide an explicit value for Ricci curvature at a given point. Another notion of lower bound on Ricci curvature, valid for length spaces equipped with a measure, has been simultaneously introduced by Sturm [46], Lott and Villani [31], and Ohta [37] (see also [43] and [41]). It relies on ideas from optimal transportation theory and analysis of paths in the space of probability measures. Their definition keeps a lot of the properties traditionally associated with positive Ricci curvature, and is compatible with the Bakry–Émery extension. However, it has two main drawbacks. First, the definition is rather involved and difficult to check on concrete examples. Second, it is infinitesimal, and difficult to adapt to discrete settings [10]. Related work. After having written a first version of this text, we...

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...Ricf = λg for some constant λ is exactly the gradient Ricci soliton equation, which plays an important role in the theory of Ricci flow. Moreover Ricf has a natural extension to metric measure spaces =-=[15, 26, 27]-=-. When f is a constant function, the Bakry-Emery Ricci tensor is the Ricci tensor so it is natural to investigate what geometric and topological results for Ricci tensor extend to the Bakry-Emery Ricc...

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.... Further related topologies on particular subspaces of isometry classes of complete and separable metric spaces have already been considered in4 ANDREAS GREVEN, PETER PFAFFELHUBER, AND ANITA WINTER =-=[Stu06]-=- and [EW06]. Convergence in these two topologies implies convergence in the Gromov-weak topology but not vice versa. Outline. The rest of the paper is organized as follows. In the next two sections we...

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...R; a closely related definition in this case was given independently by Sturm [13]. In a recent contribution, Sturm has suggested a notion of (X, d, ν) having N-Ricci curvature bounded below by K ∈ R =-=[14]-=-. These notions are preserved by measured Gromov-Hausdorff limits; when specialized to Riemannian manifolds, they coincide with classical Ricci curvature bounds. Several results in Riemannian geometry...

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...µ − 1 2 F 2 1 ∗2 W (µ,Φ) − FW 2 (µ,α) Recall that the relative entropy Ent(µ) of µ ∈ P(M) is defined by ∫ Ent(µ) := ρlog ρdm ∈ (−∞, ∞] M if µ = ρm ∈ Pac(M), and by Ent(µ) := ∞ otherwise. According to =-=[St2]-=- and [LV1], we say that (M,F,m) satisfies the curvature-dimension condition CD(K, ∞) for some K ∈ R if the relative entropy is K-convex in the sense that any µ,ν ∈ P(M) admit a minimal geodesic (µt) t...

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...y mentioning that, for general metric measure spaces, a different notion of lower bounds on the Ricci tensor based on the theory of optimal transport has been recently proposed independently by Sturm =-=[53]-=-, [54], and by Lott-Villani [40], see also the paper by Ollivier [43]. However, as pointed out by Juillet in [34], the remarkable theory developed in these papers does not appear to be suited for sub-...