The Heat Equation Under The Ricci Flow

This paper has two main results. The first deals with determining gradient estimates for positive solutions of the heat equation on a manifold whose metric is evolving under the Ricci flow. These are Li-Yau type gradient estimate, and, as an application, Harnack inequalities are given. We consider both the case when the manifold is complete and when it is compact. The second result consists of an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold of dimension at least 3, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding theorem. Considering the case when the scalar curvature is positive throughout the manifold, at any time, we will obtain, as a corollary, a bound similar to the one known for the fixed metric case