Ricci curvature for metric-measure spaces via optimal transport

By J. Lott, Cédric Villani. 2004

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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 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 inmum 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 GromovHausdor topology on compact metric spaces (modulo isometries); they form

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

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Citations in the corpus

On the geometry of metric measure spaces. II (2006, background): “Independently of [19] and [21], the particular cases CD(K,∞) and CD(0, N) of the curvature-dimension condition—essentially defined in the same form—were also discussed in a recent paper [13] by John Lott and Cédric Villani.”
STABILITY OF A 4TH-ORDER CURVATURE CONDITION ARISING IN OPTIMAL TRANSPORT THEORY (2008, methodology): “This work is a prolongation of my collaborations with John Lott [ 16 ] on the one hand, and Gr´egoire Loeper [15] on the other hand.”
The Monge Problem for Distance Cost in Geodesic Spaces (2011, background): “The basic references for this sections are [13] and [16, 17].”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “It was later adopted and slightly modified by Lott & Villani, see also the elaborate presentation in the monograph [40].”
A spectral notion of Gromov–Wasserstein distance and related methods (2011, methodology): “By exploiting the connections of Ricci curvature with mass transportation concepts, the work of Lott and Villani [49], Sturm [50] and Ollivier [51] has expanded considerably the understanding of how curvature may be defined on metric measure spaces and how it may play a role in guaranteeing compactn”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “ement of RCD(K,∞) m.m.s. which were ﬁrst introduced and studied in [AGS14] (see also [AGMR15]). ForK ∈ R,N ∈ [1,∞], the classof RCD(K,N) spacesis a subclass of CD(K,N) spacespioneered by Lott-Villani [LV09] and Sturm [Stu06a, Stu06b] a few years earlier. Roughly, RCD(K,N) spaces are those CD(K,N) sp”
Optimal maps in essentially non-branching spaces (2016, background): “Here we briefly recall the synthetic notions of lower Ricci curvature bounds, for more detail we refer to [2, 14, 21, 22, 23].”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, background): “It is by nowwell-known that, RicMn ≥ K is equivalent to metric measure space, (M, dg , dvolg), satisfying CD(K, n) curvature-dimension conditions in the sense of Lott-Sturm-Villani (see the seminal papers [30,37, 38]).”
Bochner-Weitzenb\"ock formula and Li-Yau estimates on Finsler manifolds (2011, background): “ent to Lott, Villani and the second author’s curvature-dimension condition CD(K,N). This extends the corresponding result on (weighted) Riemannian manifolds (due to [vRS], [St1], [St2], [St3], [LV2], [LV3]), and has many analytic and geometric applications (see [Oh1], [OS1] or a survey [Oh2]). Remar”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42, 72,102], as well as applications in physics [38,75,97],”
Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$CD(K,∞)-spaces (2012, methodology): “Another corollary of our result is that the RCD(K,∞) condition implies the formulation of CD(K,∞) that was used by Lott and Villani [12].”
Li–Yau inequality on finite graphs via non-linear curvature dimension conditions (2014, neutral): “[17, 18, 19, 21]).”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, neutral): “D G ] 1 4 Se p 20 07”