Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Sharp Li-Yau gradient bound on Compact Manifolds   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. In the
noncompact case one can not take $\alpha=1$ even for the hyperbolic space.

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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: econ