Hamilton's gradient estimate for the heat kernel on complete manifolds
Author:
Brett L. Kotschwar
Journal:
Proc. Amer. Math. Soc. 135 (2007), 30133019
MSC (2000):
Primary 58J35; Secondary 35K05
Published electronically:
May 14, 2007
MathSciNet review:
2317980
Fulltext PDF Free Access
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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, noncompact manifolds with . We accomplish this extension via a maximum principle of L. Karp and P. Li and a Bersteintype 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 nonnegative Ricci curvature that is sharp in the order of for the heat kernel on .
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 Hamilton, Richard S. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1 (1993), no.1, 113126. MR 1230276 (94g:58215)
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 Karp, L. and Li, P. The heat equation on complete riemannian manifolds. Unpublished notes, 1982.
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 Kuang, S. and Zhang, Q. S. A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow. arXiv:math/0611298
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 Li, Peter and Yau, S.T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 34, 153201. MR 0834612 (87f:58156)
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 Ni, Lei. A note on Perelman's LYH inequality. arXiv:math.DG/0602337
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 Ni, Lei. The entropy formula for linear heat equation. J. Geom Anal. 14 (2004), no. 1, 87100.MR 2030576 (2004m:53118a)
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 Ni, Lei and Tam, L. F. KählerRicci flow and the PoincarèLelong equation. Comm. Anal. Geom 12 (2004), no. 12, 111141.MR 2074873 (2005f:53108)
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 Perelman, Grisha. The entropy formula for the Ricci flow and its Geometric Applications. arXiv:math.DG/0211159
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 Zhang, Q. S. Some gradient estimates for the heat equation and for an equation by Perelman. Int. Math. Res. Not. 2006, Art. 1D 92314, 39 pp. MR 2250008 (2007f:35116)
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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:
http://dx.doi.org/10.1090/S0002993907088375
PII:
S 00029939(07)088375
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.
