Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

2012/09/30 by Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré · 1 citation
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paper · pdf · doi:10.1214/14-aop907

published as Annals of Probability 2015, Vol. 43, No. 1, 339-404 · Published in at http://dx.doi.org/10.1214/14-AOP907 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2014/11/12 · arxiv created 2015/01/16 · arxiv updated 2015/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The aim of the present paper is to bridge the gap between the Bakry–Émery 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 E admitting a Carré du champ Γ in a Polish measure space (X,\mathfrakm) and a canonical distance d_E that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy induced by d_E and where every function f with Γ(f)≤1 admits a continuous representative. In such a class, we show that if E satisfies a suitable weak form of the Bakry–Émery curvature dimension condition BE (K,∞) then the metric measure space (X,d,\mathfrakm) satisfies the Riemannian Ricci curvature bound RCD (K,∞) according to [Duke Math. J. 163 (2014) 1405–1490], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry–Émery BE (K,N) condition (and thus the corresponding one for RCD (K,∞) spaces without assuming nonbranching) and the stability of BE (K,N) with respect to Sturm–Gromov–Hausdorff convergence.

Citations

• A Course in Metric Geometry
• Ricci curvature for metric-measure spaces via optimal transport
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
• On the differential structure of metric measure spaces and applications
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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