A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

TL;DR

This paper simplifies the proof of sharp Li-Yau estimates on closed manifolds with negative Ricci bounds by using a maximum principle approach and extends these estimates to heat equations under Ricci flow.

Contribution

It provides a direct maximum principle proof for Li-Yau estimates and applies this method to heat equations under Ricci flow, improving understanding of heat behavior on such manifolds.

Findings

Simplified proof of Li-Yau estimates using maximum principle

Extension of estimates to heat equations under Ricci flow

Achieved Li-Yau type estimates with optimal coefficients

Abstract

Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.