Perelman's entropy and doubling property on Riemannian manifolds

TL;DR

This paper explores how certain monotone functionals of the heat kernel on Riemannian manifolds with nonnegative Ricci curvature are interconnected, showing that key geometric inequalities follow from a unified entropy inequality.

Contribution

It demonstrates that Li-Yau and Ni gradient estimates, Perelman's entropy monotonicity, and volume doubling are all derived from a single entropy inequality, unifying several fundamental results.

Findings

Gradient estimates follow from the entropy inequality

Monotonicity of Perelman's entropy is established

Volume doubling property is derived from the entropy inequality

Abstract

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volume doubling property are all consequences of an entropy inequality recently discovered by Baudoin-Garofalo. The latter is a linearized version of a logarithmic Sobolev inequality that is due to D. Bakry and M. Ledoux.