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 $M$ 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.