On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

TL;DR

This paper establishes new Harnack inequalities for the heat equation linked to the Witten Laplacian on Riemannian manifolds evolving under super Ricci flows, extending classical results to time-dependent geometric settings.

Contribution

It proves Li-Yau and Hamilton type Harnack inequalities for the Witten Laplacian under super Ricci flow conditions, generalizing previous static results to dynamic manifolds.

Findings

Proved Li-Yau type Harnack inequality for time-dependent Witten Laplacian.

Established Hamilton type dimension-free Harnack inequality.

Extended classical inequalities to manifolds with super Ricci flows.

Abstract

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows.