Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds

Ambrosio Luigi - Gigli Nicola - Savaré Giuseppe

year: 2012
abstract: The aim of the present paper is to bridge the gap between the Bakry-Emery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form $\mathcal E$ admitting a Carre' du champ $\Gamma$ in a Polish measure space $(X,d,m)$ and a canonical distance $d_{\mathcal E}$ that induces the original topology of $X$. We first characterize the distinguished class of Riemannian Energy measure spaces, where $\mathcal E$ coincides with the Cheeger energy induced by $d_{\mathcal E}$ and where every function $f$ with $\Gamma(f) \leq 1$ admits a continuous representative. In such a class we show that if $\mathcal E$ satisfies a suitable weak form of the Bakry-Emery curvature dimension condition $BE(K,\infty)$ then the metric measure space $(X,d,m)$ satisfies the Riemannian Ricci curvature bound $RCD(K,\infty)$ according to [5], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Emery condition $BE(K,N)$ (and thus the corresponding one for $RCD(K,\infty)$ spaces without assuming nonbranching) and the stability of $BE(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.

The paper is available on the cvgmt preprint server.