An improved Hamilton matrix estimates for the heat equation

TL;DR

This paper extends Hamilton's matrix Harnack estimate for the heat equation on closed manifolds by removing a curvature gradient assumption, using new mathematical tools and integral techniques.

Contribution

It provides a generalized estimate for the heat equation on all closed manifolds, answering a longstanding open question from the 1990s.

Findings

Removed the Ricci curvature gradient assumption

Established a new Hamilton matrix estimate

Utilized sharp Li-Yau estimate and tensor algebra

Abstract

In this paper, we remove the assumption on the gradient of the Ricci curvature in Hamilton's matrix Harnack estimate for the heat equation on all closed manifolds, answering a question which has been around since the 1990s. New ingredients include a recent sharp Li-Yau estimate, construction of a suitable vector field and various use of integral arguments, iteration and a little tensor algebra.