Transport inequalities, gradient estimates, entropy and Ricci curvature

Renesse, Max-K. von; Sturm, Karl-Theodor

doi:10.1002/cpa.20060

Cited by 356 publications

(271 citation statements)

References 13 publications

Supporting

Mentioning

265

Contrasting

Unclassified

Order By: Relevance

“…Recently Otto and Westdickenberg [22] have introduced techniques that were further developed by Daneri and Savare [11] to show geodesic convexity of functionals on manifolds using a purely local, Eulerian framework. The connection between bounds on displacement convexity and bounds on Ricci curvature has found important applications, see papers by Lott and Villani [16], Sturm and von Renesse [25], Sturm [24,23] and references therein. Let us also mention important applications to geometric flows, where convexity along optimal transportation paths is considered on manifolds whose metric is being deformed by a geometric flow, for example the flow by Ricci curvature.…”

Section: Introductionmentioning

confidence: 99%

Example of a displacement convex functional of first order

Carrillo

Slepčev

2009

Calc. Var.

View full text Add to dashboard Cite

Abstract. We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.

show abstract

Section: Introductionmentioning

confidence: 99%

Example of a displacement convex functional of first order

Carrillo

Slepčev

2009

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…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%

“…This kind of condition appears in [RS05] as one among several ways to characterize a lower Ricci bound on manifolds. It was proposed in [Oll07, Oll09] (see also [Jou07]) as a possible definition of Ricci curvature extending to metric spaces several results known for positively curved manifolds.…”

Section: Nmentioning

confidence: 99%

See 1 more Smart Citation

A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

View full text Add to dashboard Cite

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.

show abstract

“…[1,2,3,4,10,11,14,15,16,17,22,25,26,28,29,30,32,33,34,35] and the references therein for a very noncomprehensive excerpt). In the study of n-dimensional weighted-manifolds satisfying CD(ρ, N ), the range of admissible values for N has traditionally been N ∈ [n, ∞].…”

Section: Introductionmentioning

confidence: 99%

Harmonic Measures on the Sphere via Curvature-Dimension

Milman¹

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

View full text Add to dashboard Cite

Transport inequalities, gradient estimates, entropy and Ricci curvature

Cited by 356 publications

References 13 publications

Example of a displacement convex functional of first order

Example of a displacement convex functional of first order

A visual introduction to Riemannian curvatures and some discrete generalizations

Harmonic Measures on the Sphere via Curvature-Dimension

Contact Info

Product

Resources

About