A curvature-dimension condition for metric measure spaces

Abstract

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 ⋅ | ξ | 2 for all ξ ∈ 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 , ∞ ) . 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. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 342 (2006).