A direct approach to sharp Li-Yau estimates on closed manifolds with negative Ricci lower bound (Q6654042)

In this paper, the authors simplify Zhang's proof [\textit{Q. S. Zhang}, ``A sharp Li-Yau gradient bound on compact manifolds'', Preprint, \url{arXiv:2110.08933}] of a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds of Ricci curvature bounded below by a negative constant. More precisely, Zhang proved that if \((M^n, g)\) is an \(n\)-dimensional closed Riemannian manifold with Ric \(\ge -K\) for some constant \(K\ge 0\) and \(u\) a positive solution of the heat equation\N\[\N\left(\Delta-\frac{\partial}{\partial t}\right)u=0\N\]\Non \(M\times [0,\infty)\), then there exist constants \(c_1, c_2\) depending only on \(n\) such that\N\[\Nt\left(\frac{|\nabla u|^2}{u^2}-\frac{\partial_tu}{u}\right) \le\frac{n}{2}+\sqrt{2nK(1+Kt)(1+t)\text{diam}_M}+\sqrt{M(1+Kt)(c_1+c_2K)t}\N\]\Nfor all \((x,t)\in M\times[0,\infty)\).\N\NThe main ingredient of Zhang's proof is an integral iteration argument which uses Hamilton's gradient estimate, heat kernel Gaussian bounds and the parabolic Harnack inequality. In this paper, the authors obtain the same sharp Li-Yau estimate through a classical maximum principle argument. Furthermore, the authors apply the same reasoning to the heat and conjugate heat equations under the Ricci flow, proving Li-Yau type estimates with optimal coefficients.