Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow

TL;DR

This paper develops new differential Harnack inequalities for backward heat equations with potentials under Ricci flow, including Perelman's inequality, advancing understanding of heat behavior in evolving geometries.

Contribution

It introduces a general evolution formula for Harnack quantities and proves several new inequalities, including Perelman's, for backward heat equations with potentials under Ricci flow.

Findings

Derived a general evolution formula for Harnack quantities

Proved several differential Harnack inequalities under Ricci flow

Established Perelman's Harnack inequality for conjugate heat equation

Abstract

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.