Differential Harnack inequalities on Riemannian manifolds I : linear heat equation

TL;DR

This paper develops new gradient estimates and Harnack inequalities for the heat equation on Riemannian manifolds with Ricci curvature bounded below, leading to improved heat kernel bounds and entropy formulas.

Contribution

It introduces novel Li-Yau type gradient estimates and a Perelman type differential Harnack inequality, extending previous results to broader classes of manifolds.

Findings

New Li-Yau gradient estimates for heat solutions

Perelman type Harnack inequality on manifolds with Ricci bounds

Enhanced heat kernel and entropy estimates

Abstract

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$, which generalizes a result of L. Ni \cite{NL1,NL4}. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.