An upper bound of the heat kernel along the harmonic-Ricci flow

TL;DR

This paper establishes an upper bound for the heat kernel along the harmonic-Ricci flow by deriving Sobolev inequalities and applying Moser's iteration, contributing to the understanding of heat kernel behavior in geometric flows.

Contribution

It introduces a new upper bound estimate for the heat kernel along the harmonic-Ricci flow using Sobolev inequalities and parabolic estimates.

Findings

Derived a Sobolev inequality along the harmonic-Ricci flow.

Proved a linear parabolic estimate based on the Sobolev inequality.

Obtained an upper bound estimate for the heat kernel under the flow.

Abstract

In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.