RICCI CURVATURE FOR METRIC-MEASURE SPACES VIA OPTIMAL TRANSPORT

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 P2(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. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having “curvature bounded below by K”, with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the Gromov–Hausdorff topology on compact metric spaces (modulo isometries); they form a closed subset. In view of Alexandrov’s work, it is natural to ask whether there are metric space versions of other types of Riemannian curvature, such as Ricci curvature. This question takes substance from Gromov’s precompactness theorem for Riemannian manifolds with Ricci curvature bounded below by K, dimension bounded above by N and diameter bounded above by D [23, Theorem 5.3]. The precompactness indicates that there could be a notion of a length space having “Ricci curvature bounded below by K”, special cases of which would be Gromov–Hausdorff limits of manifolds with lower Ricci curvature bounds.  2006. The research of the first author was supported by NSF grant DMS-0306242 and the Clay Mathematics Institute. 1 2 JOHN LOTT AND CÉDRIC VILLANI Gromov–Hausdorff limits of manifolds with Ricci curvature bounded below have been studied by various authors, notably Cheeger and Colding [15, 16, 17, 18]. One feature of their work, along with the earlier work of Fukaya [21], is that it turns out to be useful to add an auxiliary Borel probability measure ν and consider metric-measure spaces (X, d, ν). (A compact Riemannian manifold M has a canonical measure ν given by the normalized Riemannian density dvolM vol(M) .) There is a measured Gromov–Hausdorff topology on such triples (X, d, ν) (modulo isometries) and one again has precompactness for Riemannian manifolds with Ricci curvature bounded below by K, dimension bounded above by N and diameter bounded above by D. Hence the question is whether there is a good notion of a measured length space (X, d, ν) having “Ricci curvature bounded below by K”. Whatever definition one takes, one would like the set of such triples to be closed in the measured Gromov–Hausdorff topology. One would also like to derive some nontrivial consequences from the definition, and of course in the case of Riemannian manifolds one would like to recover classical notions. We refer to [16, Appendix 2] for further discussion of the problem of giving a “synthetic” treatment of Ricci curvature. Our approach is in terms of a metric space (P (X),W2) that is canonically associated to the original metric space (X, d). Here P (X) is the space of Borel probability measures on X and W2 is the so-called Wasserstein distance of order 2. The square of the Wasserstein distance W2(μ0, μ1) between μ0, μ1 ∈ P (X) is defined to be the infimal cost to transport the total mass from the measure μ0 to the measure μ1, where the cost to transport a unit of mass between points x0, x1 ∈ X is taken to be d(x0, x1). A transportation scheme with infimal cost is called an optimal transport. The topology on P (X) coming from the metric W2 turns out to be the weak-∗ topology. We will write P2(X) for the metric space (P (X),W2), which we call the Wasserstein space. If (X, d) is a length space then P2(X) turns out to also be a length space. Its geodesics will be called Wasserstein geodesics. If M is a Riemannian manifold then we write P ac 2 (M) for the elements of P2(M) that are absolutely continuous with respect to the Riemannian density. In the past fifteen years, optimal transport of measures has been extensively studied in the case X = R, with motivation coming from the study of certain partial differential equations. A notion which has proved useful is that of displacement convexity, i.e. convexity along Wasserstein geodesics, which was introduced by McCann in order to show the existence and uniqueness of minimizers for certain relevant functions on P ac 2 (R ) . In the past few years, some regularity results for optimal transport on R have been extended to Riemannian manifolds [19, 32]. This made it possible to study displacement convexity in a Riemannian setting. Otto and Villani  carried out Hessian computations for certain functions on P2(M) using a formal infinite-dimensional Riemannian structure on P2(M) defined by Otto [35]. These formal computations indicated a relationship between the Hessian of an “entropy” function on P2(M) and the Ricci curvature of M . Later, a rigorous displacement convexity result for a class of functions on P ac 2 (M), when M has nonnegative Ricci curvature, was proven by Cordero-Erausquin, McCann and Schmuckenschläger . This work was extended by von Renesse and Sturm [40]. RICCI CURVATURE VIA OPTIMAL TRANSPORT 3 Again in the case of Riemannian manifolds, a further circle of ideas relates displacement convexity to log Sobolev inequalities, Poincaré inequalities, Talagrand inequalities and concentration of measure [8, 9, 27, 36]. In this paper we use optimal transport and displacement convexity in order to define a notion of a measured length space (X, d, ν) having Ricci curvature bounded below. If N is a finite parameter (playing the role of a dimension) then we will define a notion of (X, d, ν) having nonnegative N -Ricci curvature. We will also define a notion of (X, d, ν) having ∞-Ricci curvature bounded below by K ∈ R. (The need to input the possibly-infinite parameter N can be seen from the Bishop–Gromov inequality for complete n-dimensional Riemannian manifolds with nonnegative Ricci curvature. It states that r vol(Br(m)) is nonincreasing in r, where Br(m) is the r-ball centered at m [23, Lemma 5.3.bis]. When we go from manifolds to length spaces there is no a priori value for the parameter n. This indicates the need to specify a dimension parameter in the definition of Ricci curvature bounds.) We now give the main results of the paper, sometimes in a simplified form. For consistency, we assume in the body of the paper that the relevant length space X is compact. The necessary modifications to deal with complete pointed locally compact length spaces are given in Appendix E. Let U : [0,∞) → R be a continuous convex function with U(0) = 0. Given a reference probability measure ν ∈ P (X), define the function Uν : P2(X) → R ∪ {∞} by