A curvature-dimension condition for metric measure spaces

Sturm, Karl-Theodor

doi:10.1016/j.crma.2005.11.008

Probability Theory

A curvature-dimension condition for metric measure spaces
[Une condition de type courbure-dimension pour des espaces métriques mesurés]

Présenté par : Paul Malliavin

Sturm, Karl-Theodor ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Tome 342 (2006) no. 3, pp. 197-200.

Résumé
Abstract

Nous présentons une condition de type courbure-dimension $CD (K, N)$ pour des espaces métriques mesurés $(M, d, m)$ , qui peut être considérée comme une contrepartie géométrique de celle de Bakry–Émery [D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [1]] pour les formes de Dirichlet. Pour les variétés riemanniennes, elle est satisfaite si et seulement si $\dim (M) ⩽ N$ et ${Ric}_{M} (ξ, ξ) ⩾ K \cdot {| ξ |}^{2}$ pour tout $ξ \in TM$ . La borne de la courbure [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., in press. [4] ; K.T. Sturm, Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 235–238. [6] ; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press. [7]] est le cas limite $CD (K, \infty)$ .

Notre condition est stable pour la convergence. Elle comporte des conséquences géométriques diverses, comme les théorèmes de Bishop–Gromov et de Bonnet–Myers. Dans les deux cas, on obtient des estimations optimales connues dans le cas riemannien.

We present a curvature-dimension condition $CD (K, N)$ for metric measure spaces $(M, d, m)$ . In some sense, it will be the geometric counterpart to the Bakry–Émery [D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [1]] condition for Dirichlet forms. For Riemannian manifolds, it holds if and only if $\dim (M) ⩽ N$ and ${Ric}_{M} (ξ, ξ) ⩾ K \cdot {| ξ |}^{2}$ for all $ξ \in TM$ . The curvature bound introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., in press. [4]; K.T. Sturm, Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 235–238. [6]; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press. [7]] is the limit case $CD (K, \infty)$ .

Our curvature-dimension condition is stable under convergence. Furthermore, it entails various geometric consequences e.g. the Bishop–Gromov theorem and the Bonnet–Myers theorem. In both cases, we obtain the sharp estimates known from the Riemannian case.

Reçu le : 2005-06-03
Accepté le : 2005-11-07
Publié le : 2005-12-20

DOI : 10.1016/j.crma.2005.11.008

Affiliations des auteurs :

Sturm, Karl-Theodor ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany

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Sturm, Karl-Theodor. A curvature-dimension condition for metric measure spaces. Comptes Rendus. Mathématique, Tome 342 (2006) no. 3, pp. 197-200. doi : 10.1016/j.crma.2005.11.008. http://www.numdam.org/articles/10.1016/j.crma.2005.11.008/

Bibliographie
Cité par

[1] Bakry, D.; Émery, M. Diffusions hypercontractives, Séminaire de Probabilités XIX, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206

[2] Bakry, D.; Qian, Z. Some new results on eigenvectors via dimension, diameter and Ricci curvature, Adv. in Math., Volume 155 (2000), pp. 98-153

[3] Cordero-Erausquin, D.; McCann, R.; Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascamb and Lieb, Invent. Math., Volume 146 (2001), pp. 219-257

[4] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Preprint, 2004

[5] Sturm, K.T. Convex functionals of probability measures and nonlinear diffusions on manifolds, J. Math. Pures Appl., Volume 84 (2005), pp. 149-168

[6] Sturm, K.T. Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 235-238

[7] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press

[8] K.T. Sturm, On the geometry of metric measure spaces. II, Acta Math., in press

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