Davies type estimate and the heat kernel bound under the Ricci flow

TL;DR

This paper establishes a Davies type double integral estimate for the heat kernel under Ricci flow, confirming a prior question and offering a new proof for Gaussian bounds on the heat kernel.

Contribution

It introduces a Davies type estimate for the heat kernel under Ricci flow and applies it to derive Gaussian bounds, advancing understanding of heat kernel behavior.

Findings

Proved a Davies type double integral estimate for the heat kernel.

Confirmed a question posed by Chow et al. regarding heat kernel estimates.

Provided a new proof of Gaussian upper and lower bounds for the heat kernel.

Abstract

We prove a Davies type double integral estimate for the heat kernel $H(y,t;x,l)$ under the Ricci flow. As a result, we give an affirmative answer to a question proposed by Chow etc.. Moreover, we apply the Davies type estimate to provide a new proof of the Gaussian upper and lower bounds of $H(y,t;x,l)$ which were first shown by Chau-Tam-Yu.