Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero–Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael

doi:10.5802/afst.1132

Cited by 58 publications

(85 citation statements)

References 34 publications

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“…In particular, these authors prove the implication (1) ⇒ (4) of Theorem 7.3(a) of the present paper when Ψ is constant and N = n. The paper [19] was extended by von Renesse and Sturm [40], whose paper contains a proof of the implications (1) ⇔ (5) of Theorem 7.3(b) of the present paper when Ψ is constant, and also indicates that the condition (5) may make sense for some metric-measure spaces. In a more recent contribution, which was done independently of the present paper, Cordero-Erausquin, McCann and Schmuckenschläger [20] prove the implication (1) ⇒ (5) of Theorem 7.3(b) for general Ψ.…”

Section: Appendix a The Wasserstein Space As An Alexandrov Spacementioning

confidence: 59%

Ricci curvature for metric-measure spaces via optimal transport

2009

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Abstract. 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 P 2 (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. 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.) There is a measured Gromov-Hausdorff topology on such triples (X, d, ν) (modulo isom...

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Section: Appendix a The Wasserstein Space As An Alexandrov Spacementioning

confidence: 59%

Ricci curvature for metric-measure spaces via optimal transport

2009

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“…but exploiting the contracted second Bianchi identity 2d Ric þ dR ¼ 0 (using the notation and conventions of [16] We are now in a position to compute the volume element aðtÞ of the lemma, in the spirit of classical comparison geometry, and following the analogous [7], Lemma 6. By definition, aðtÞ ¼ Àln det A, and so…”

Section: Behaviour Of Boltzmann-shannon Entropy Along L-wasserstein Gmentioning

confidence: 99%

“…But following [7], We are now in a position to investigate the behaviour of this entropy along L-Wasserstein geodesics (as defined in the previous section) with a result analogous to [11] The j of the lemma is the j which induces the L-Wasserstein geodesic under consideration. This also induces a map F via (2.5) which we use below.…”

Section: Behaviour Of Boltzmann-shannon Entropy Along L-wasserstein Gmentioning

confidence: 99%

ℒ-optimal transportation for Ricci flow

Topping¹

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The nineteenth-century-old Brunn-Minkowski inequality (see e.g. [Gar02] for an introduction) can be seen in geodesic spaces as a weaker version of CD(K, N ), using a larger set of midpoints A 1/2 ; the equivalence with Ricci curvature in the Riemannian case is proven in [CMS06,CMS01]. As a general property of geodesic spaces, it is weaker than CD(K, N ) but already has a number of the same geometrical consequences [Stu06].…”

Section: Nmentioning

confidence: 99%

“…The reverse is true in negative Ricci curvature. This intuition, illustrated in Figures 11 and 12, was formalized in various ways in [CMS01,RS05,CMS06].…”

Section: Nmentioning

confidence: 99%

A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

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Abstract. We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.The first part of this text covers in a hopefully intuitive and visual way some of the usual objects of Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, the Riemann tensor, Bianchi identities... For each of those we try to provide one or several pictures that convey the meaning (or at least one possible interpretation) of the formal definition.Next, we consider the problem of defining analogues of curvature for nonsmooth or discrete objets. This is in the spirit of "synthetic" or "coarse" geometry, in which large-scale properties of metric spaces are investigated instead of fine smallscale properties; one of the aims being to be impervious to small perturbations of the underlying space. Motivation for these non-smooth or discrete extensions often comes from various other fields of mathematics, such as group theory or optimal transport. From this viewpoint is emerging a theory of metric measure spaces, see for instance [Gro99].Thus we cover the definitions and mention some applications of δ-hyperbolic spaces and Alexandrov curvature (generalizing sectional curvature), and of two notions of generalized Ricci curvature: displacement convexity of entropy introduced by Sturm and Lott-Villani (following Renesse-Sturm and others), and coarse Ricci curvature as used by the author. We mention new theorems obtained for the original Riemannian case thanks to this more general viewpoint. We also briefly discuss the differences between these two approaches to Ricci curvature.Note that scalar curvature is not covered, for lack of the author's competence about recent developments and applications in this field.

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Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cited by 58 publications

References 34 publications

Ricci curvature for metric-measure spaces via optimal transport

Ricci curvature for metric-measure spaces via optimal transport

ℒ-optimal transportation for Ricci flow

A visual introduction to Riemannian curvatures and some discrete generalizations

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