Bounds on the Heat Kernel under the Ricci Flow

TL;DR

This paper derives bounds on the heat kernel for manifolds evolving under Ricci flow, extending known fixed metric results and depending on Sobolev embedding constants, with special cases for positive scalar curvature.

Contribution

It provides new heat kernel estimates under Ricci flow, incorporating Sobolev constants and positive scalar curvature conditions, advancing understanding of geometric analysis during evolution.

Findings

Established heat kernel bounds depending on Sobolev constants

Extended fixed metric heat kernel bounds to Ricci flow scenarios

Derived specific bounds for manifolds with positive scalar curvature

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.