Sectional and intermediate Ricci curvature lower bounds via Optimal Transport

TL;DR

This paper provides a novel characterization of sectional and p-Ricci curvature bounds in Riemannian manifolds using optimal transport and entropy convexity, leading to new geometric inequalities.

Contribution

It introduces an optimal transport framework to characterize curvature bounds via entropy convexity, extending to p-Ricci curvature and deriving new inequalities.

Findings

Characterization of sectional curvature bounds through entropy convexity.

Extension to p-Ricci curvature and trace-based bounds.

New Brunn-Minkowski type inequality involving p-dimensional measures.

Abstract

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.