On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

Matthias Erbar; Kazumasa Kuwada; Karl Theodor Sturm

doi:10.1007/s00222-014-0563-7

On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

Matthias Erbar, Kazumasa Kuwada, Karl Theodor Sturm

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

124 Citations (Scopus)

Abstract

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$^L2-Wasserstein distance.

Original language	English
Pages (from-to)	993-1071
Number of pages	79
Journal	Inventiones Mathematicae
Volume	201
Issue number	3
DOIs	https://doi.org/10.1007/s00222-014-0563-7
Publication status	Published - 2015 Sep 22

ASJC Scopus subject areas

Mathematics(all)

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AU - Kuwada, Kazumasa

AU - Sturm, Karl Theodor

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Y1 - 2015/9/22

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AB - We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$L2-Wasserstein distance.

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On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

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