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

DOI:
https://doi.org/10.1090/S0002-9939-07-08837-5

Published electronically:
May 14, 2007

MathSciNet review:
2317980

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Abstract | References | Similar Articles | Additional Information

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 . 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 for the heat kernel on .

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Additional Information

**Brett L. Kotschwar**

Affiliation:
Department of Mathematics, University of California, San Diego, California 92110

Email:
bkotschw@math.ucsd.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-08837-5

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.