Bounds on the heat kernel under the Ricci flow

Abstract

We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold M M of dimension at least 3 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.