Gradient estimates for the heat equation under the Ricci-Harmonic Map flow

TL;DR

This paper derives gradient estimates and Harnack inequalities for the heat equation on manifolds evolving under Ricci and harmonic map flows, extending classical results to dynamic geometric settings.

Contribution

It provides new gradient estimates and Li-Yau type inequalities for heat equations on manifolds under coupled Ricci and harmonic map flows, a novel extension of existing static results.

Findings

Established gradient estimates for heat solutions under Ricci-harmonic flow

Proved Li-Yau type Harnack inequalities in this setting

Applicable to both complete and compact manifolds without boundary

Abstract

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled with the harmonic map flow between $M$ and a second manifold $N$. We prove Li-Yau type Harnack inequalities and we consider the cases when $M$ is a complete manifold without boundary and when $M$ is compact, without boundary.