# Bochner's inequality and the entropic curvature dimension condition for metric measure spaces

## Matthias Erbar

created by paolini on 25 Apr 2014
modified by gigli on 03 Jun 2014

Abstract.

(joint work with K. Kuwada and K.Th. Sturm) Bochner's inequality is one of the fundamental estimates in geometric analysis on Riemannian manifolds. It states that

$$\frac12\Delta \nabla u 2-\langle\nabla u, \nabla\Delta u\rangle\ge K \cdot \nabla u 2+\frac1N \cdot \Delta u 2$$

for each smooth function $u$ on a Riemannian manifold provided K is a lower bound for the Ricci curvature on and N is an upper bound for the dimension.

The main result I present in this talk is a Bochner inequality on infinitesimally Hilbertian metric measure spaces satisfying the (reduced) curvature-dimension condition. Moreover, also the converse is true: an appropriate version of the Bochner inequality (for the canonical gradient and Laplacian) on an infinitesimally Hilbertian mms will imply the reduced curvature-dimension condition. Our approach relies on the new so-called entropic curvature-dimension condition which encodes bounds on the curvature and dimension through refined convexity properties of the Boltzmann entropy (and not the Renyi entropy). Besides that, I will present also new, sharp Wasserstein-contraction results for the heat flow as well as Bakry-Ledoux type gradient estimates for the heat semigroup, each of which are equivalent to the (reduced) curvature-dimension condition.

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