A Gaussian upper bound of the conjugate heat equation along an extended Ricci flow

TL;DR

This paper establishes a Gaussian upper bound for the conjugate heat equation's fundamental solutions along an extended Ricci flow, advancing understanding of heat distribution in evolving geometric spaces.

Contribution

It introduces a Sobolev inequality along the extended Ricci flow and proves a Gaussian upper bound for the conjugate heat equation's fundamental solutions, which is a novel result.

Findings

Derived a Sobolev inequality along the extended Ricci flow.

Proved a Gaussian upper bound for the fundamental solutions of the conjugate heat equation.

Enhanced understanding of heat behavior in evolving geometric structures.

Abstract

In this paper, we derive a Sobolev inequality along an extended Ricci flow and prove a point-wise Guassian type bound for the fundamental solutions of the conjugate heat equation under the flow.