A DIRECT APPROACH TO SHARP LI-YAU ESTIMATES WITH NEGATIVE RICCI LOWER BOUND

Recently, Qi S. Zhang [ A Sharp Li-Yau gradient bound on Compact Manifolds, https://arxiv.org/2110.08933, 2021] has derived a sharp LiYau 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 Qi S. Zhang's paper. 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.