Hamilton differential Harnack inequality and $W$-entropy for Witten Laplacian on Riemannian manifolds

TL;DR

This paper establishes a Hamilton differential Harnack inequality and introduces a W-entropy formula for the Witten Laplacian on Riemannian manifolds under certain curvature conditions, advancing understanding of heat equations in geometric analysis.

Contribution

It proves the Hamilton differential Harnack inequality and formulates the W-entropy for the Witten Laplacian on manifolds satisfying the CD(-K, m) condition, extending previous results to broader geometric contexts.

Findings

Proved Hamilton differential Harnack inequality for Witten Laplacian.

Derived W-entropy formula for fundamental solutions.

Extended results to manifolds with super Ricci flows.

Abstract

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition and on compact manifolds equipped with $(-K, m)$-super Ricci flows.