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

Author: Brett L. Kotschwar
Journal: Proc. Amer. Math. Soc. 135 (2007), 3013-3019
MSC (2000): Primary 58J35; Secondary 35K05
Published electronically: May 14, 2007
MathSciNet review: 2317980
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Abstract: 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 Berstein-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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Additional Information

Brett L. Kotschwar
Affiliation: Department of Mathematics, University of California, San Diego, California 92110

Received by editor(s): March 13, 2006
Received by editor(s) in revised form: June 23, 2006
Published electronically: May 14, 2007
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.