Hamilton's gradient estimate for the heat kernel on complete manifolds

Details Hamilton's gradient estimate for the heat kernel on complete manifolds Hamilton's gradient estimate for the heat kernel on complete manifolds

2007-01-12

arxiv.org/abs/math/0701335v1

Mathematics

Analysis of PDEs

6 pages, to appear in Proc. Amer. Math. Soc

Scientific paper

In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish this extension via a maximum principle of L. Karp and P. Li and a Bernstein-type estimate on the gradient of the solution. An application of our result, together with the bounds of P. Li and S.T. Yau, yields an estimate on the gradient of the heat kernel for complete manifolds with non-negative Ricci curvature that is sharp in the order of $t$ for the heat kernel on ${\mathbb{R}}^n$.

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