Gaussian upper bounds for the heat kernel on evolving manifolds

TL;DR

This paper establishes Gaussian upper bounds for the heat kernel on manifolds evolving under geometric flows, using logarithmic Sobolev inequalities and ultracontractivity, extending known results to dynamic settings.

Contribution

It introduces a flexible method to derive heat kernel bounds on evolving manifolds, including new proofs for Ricci flow cases with curvature bounds.

Findings

Gaussian upper bounds for heat kernels on evolving manifolds

New proofs for Ricci flow heat kernel bounds

Extension to other geometric flows

Abstract

In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.