Characterization of pinched Ricci curvature by functional inequalities

TL;DR

This paper establishes functional inequalities equivalent to pinched Ricci curvature on Riemannian manifolds, extending to geometric flows, and uses these to characterize Ricci flow through such inequalities.

Contribution

It introduces new functional inequalities that characterize pinched Ricci curvature and extends these results to manifolds with geometric flows, including Ricci flow.

Findings

Functional inequalities equivalent to pinched Ricci curvature.

Gradient and $L^p$-inequalities established.

Characterization of Ricci flow via functional inequalities.

Abstract

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, $L^p$-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.