A Sharp Li-Yau gradient bound on Compact Manifolds

Abstract

Let $(\M^n, g)$ be a $n$ dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant $-K \le 0$. Let $u$ be a positive solution of the heat equation on $\M^n \times (0, \infty)$. The well known Li-Yau gradient bound states that $$ t \left(\frac{|\nabla u|^2}{u^2} - \alpha\frac{\pa_t u}{u}\right) \leq \frac{n\alpha^2}{2} + t \frac{n\alpha^2K}{2(\alpha-1)},\quad \forall \alpha>1, t>0. $$ The bound with $\alpha =1$ is sharp if $K=0$. If $-K < 0$, the bound tends to infinity if $\alpha=1$. In over 30 years, several sharpening of the bounds have been obtained with $\alpha$ replaced by several functions $\alpha=\alpha(t)>1$ but not equal to $1$. An open question (\cite{CLN}, \citeLX} etc) asks if a sharp bound can be reached. In this short note, we observe that for all complete compact manifolds one can take $\alpha=1$. Thus a sharp bound, up to computable constants, is found in the compact case. This result also seems to sharpen Theorem 1.4 in \cite{LY} for compact manifolds with convex boundaries. In the noncompact case one can not take $\alpha=1$ even for the hyperbolic space. An example is also given, which shows that there does not exist an optimal function of time only $\alpha=\alpha(t)$ for all noncompact manifolds with Ricci lower bound, giving a negative answer to the open question in the noncompact case.