Calculus of Variations and Geometric Measure Theory

C. Ketterer - A. Mondino

Sectional and intermediate Ricci curvature lower bounds via Optimal Transport

created by mondino on 11 Oct 2016
modified on 26 Feb 2018


Accepted Paper

Inserted: 11 oct 2016
Last Updated: 26 feb 2018

Journal: Advances in Mathematics
Year: 2016


The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called $p$-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role of reference measure is played by the $p$-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving $p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure.