Gradient estimates for the heat equation under the Ricci flow

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 \(M\).