Harnack Estimates for Conjugate Heat Kernel on Evolving Manifolds

TL;DR

This paper develops Harnack estimates for the conjugate heat kernel on evolving manifolds within an abstract geometric flow, unifying known results and extending them to Ricci-Harmonic and Lorentzian mean curvature flows.

Contribution

It introduces a correction term D in the Harnack estimate framework, enabling new results and unification of existing inequalities in geometric flows.

Findings

Unified Harnack inequality framework for geometric flows

Extension of results to Ricci-Harmonic flow and Lorentzian mean curvature flow

New inequalities under nonnegative sectional curvature conditions

Abstract

In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation provides a unified framework for some known results, in particular including corresponding results of Ni, Perelman, and Tran as special cases. Moreover it leads to new results in the setting of Ricci-Harmonic flow and mean curvature flow in Lorentzian manifolds with nonnegative sectional curvature.