An improvement of the sharp Li-Yau bound on closed manifolds (Q6614085)

In this paper, the author generalizes \textit{Q. S. Zhang}'s work [``A Sharp Li-Yau gradient bound on compact manifolds'', Preprint, \url{arXiv:2110.08933}] on sharp Li-Yau gradient bounds on compact manifolds.\N\NMore precisely, Zhang proved that if \((M,g)\) is an n-dimensional closed Riemannian manifold with Ricci curvature \(\text{Ric} \ge -K\) for some \(K\ge 0\) and \(u\) is a positive smooth solution of the heat equation \((\Delta-\partial_t)u=0\) on \(M\times[0,\infty)\), then there exist constants \(c_1,c_2\) depending only on \(n\) such that\N\begin{align*}\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\\\N&+\sqrt{K(1+Kt)(c_1+c_2K)t},\N\end{align*}\Nfor all \((x,t)\in M\times[0,\infty)\). The proof relies on an integral iteration argument that uses Hamilton's gradient estimates. In this paper, the author proves that, under the same assumptions, there are constants \(c_1,c_2\) depending only on \(n\) such that\N\[\Nt\left(\frac{|\nabla u|^2}{u^2}-\frac{\partial_tu}{u}\right)\le\frac{n}{2}+\sqrt{\frac{nKt}{\varphi(t/2)}(1+t)}\text{diam}_M+\sqrt{\frac{Kt}{\varphi(t/2)}(c_1+c_2K)}t,\N\]\Nfor all \((x,t)\in M\times[0,\infty)\), where \(\varphi\) is a nonnegative function such that \(\frac{t}{\varphi(t/2)}\) is nondecreasing and \(\varphi'(t)+2K\varphi(t)\le 1\), with \(\varphi(0)=0\). This generalizes Zhang's result, as the choice \(\varphi(t)=\frac{t}{1+2Kt}\) recovers it.