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- W4286901868 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 $$ t left(frac{|nabla u|^2}{u^2} - alphafrac{pa_t u}{u}right) leq frac{nalpha^2}{2} + t frac{nalpha^2K}{2(alpha-1)},quad forall alpha>1, t>0. $$ 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." @default.
- W4286901868 created "2022-07-25" @default.
- W4286901868 creator A5031096866 @default.
- W4286901868 date "2021-10-17" @default.
- W4286901868 modified "2023-09-23" @default.
- W4286901868 title "A Sharp Li-Yau gradient bound on Compact Manifolds" @default.
- W4286901868 doi "https://doi.org/10.48550/arxiv.2110.08933" @default.
- W4286901868 hasPublicationYear "2021" @default.
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