Inventiones mathematicae

, Volume 201, Issue 3, pp 993–1071 | Cite as

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

  • Matthias Erbar
  • Kazumasa Kuwada
  • Karl-Theodor Sturm


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 \(\Gamma _2\)-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 \(L^2\)-Wasserstein distance.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthias Erbar
    • 1
  • Kazumasa Kuwada
    • 2
  • Karl-Theodor Sturm
    • 1
  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany
  2. 2.Graduate School of ScienceTokyo Institute of TechnologyMeguro-kuJapan

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