Gradient estimates for the heat equation under the Ricci flow

TL;DR

This paper derives gradient estimates and Li-Yau inequalities for positive solutions of the heat equation on manifolds evolving under Ricci flow, with applications to Harnack inequalities.

Contribution

It extends gradient estimate techniques to Ricci flow settings, providing new inequalities for heat equations on evolving manifolds.

Findings

Li-Yau-type inequalities established for Ricci flow

Gradient estimates derived for heat solutions on evolving manifolds

Harnack inequalities obtained as applications

Abstract

The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where $M$ is a complete manifold without boundary and the case where $M$ is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on $M$.