Gradient estimates for the heat equation under the Ricci flow

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.