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

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2012

347 citations recorded by the source corpus.

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 E admitting a Carre 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–Emery 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–Emery 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Genetics

View Paper: https://www.semanticscholar.org/paper/04129e266c4d0a74ebefef0e8351b1bca3ce0ed1

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Citations in the corpus

Metric measure spaces with Riemannian Ricci curvature bounded from below (2011, neutral): “D G ] Contents”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “We also mention the more recent Riemannian curvature dimension condition RCD introduced in the infinite dimensional case in [3, 5, 2] and then investigated by various authors in the finite dimensional refinement.”
On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces (2013, background): “While this characterization is extremely useful in the proof of stability properties [3,4,23], in the proof of localization or globalization properties it suffers the same limitations described in Remark 2.”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, background): “We also mention the more recent Riemannian curvature dimension condition RCD∗ introduced in the infinite dimensional case in [4, 2, 1] and in the finite dimensional case in [28, 5].”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “2 is the same as what is assumed in [5] and it is always satisfied if (X, d,m) is RCD(K ′,∞) for any K ′ ∈ R.”
The globalization theorem for the Curvature-Dimension condition (2016, background): “ the inﬁnite dimensional case N= ∞, it was introduced in [7] for ﬁnite measures mand in [4] for σ-ﬁnite ones. The class RCD∗(K,N) with N<∞ has been proposed in [40] and extensively investigated in [8, 35, 11]. We refer to these papers and references therein for a general account on the synthetic ”