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

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, m) and a canonical distance dE 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 dE 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, m) 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.

Obata’s Rigidity Theorem for Metric Measure Spaces

“…For more details see for instance [1][2][3]. Let L (mX) = L (X) be the Lesbegue-space of (X, dX, mX).…”

Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

“…In the flat Euclidean case M = R d , V ≡ 0, (6) with N = ∞ simply follows by Jensen's inequality and the commutation property of the heat equation ∇P t f = P t (∇ f ). In the general case, (6) reflects the commutator bounds coded in (5); the simplest situation is provided by the BE(K, ∞) case, when (6) follows (at least formally -see, for example, [9,Sec.…”

“…In the general case, (6) reflects the commutator bounds coded in (5); the simplest situation is provided by the BE(K, ∞) case, when (6) follows (at least formally -see, for example, [9,Sec. 3.2.3]) by the monotonicity property of the quantity…”

“…Notice that one of the advantages of (15) with respect to (6) and to the differential formulation of the BE(K, ∞) condition given by (5) relies on the weaker regularity requirement of its formulation: it involves probability measures and the Markov semigroup (P t ) t≥0 but avoids local differentiability structures. This is one of the recurrent themes of pushing the geometric information coded into lower Ricci curvature bounds toward general metric measure spaces.…”

“…It turns out that these are, in fact, equivalent and can be unified by the notion of EVI K flow [4,5,6].…”