# 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.