Calculus of Variations and Geometric Measure Theory

Bochner's inequality and the entropic curvature-dimension condition on metric measure spaces

Matthias Erbar

created by dicastro on 24 Feb 2014

26 feb 2014 -- 17:00   [open in google calendar]

Aula Riunioni - Department of Mathematics, University of Pisa

Abstract.

Bochner's inequality is one of the most 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 metric measure spaces with linear heat flow and satisfying the (reduced) curvature-dimension condition. Indeed, also the converse is true: if the heat flow on a metric measure space is linear then an appropriate version of the Bochner inequality (for the canonical gradient and Laplacian) will imply the reduced curvature-dimension condition. Besides that, I will present also new, sharp Wasserstein-contraction results for the heat flow as well as Bakry-Ledoux type gradient estimates each of which is equivalent to the curvature-dimension condition.