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 | 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 .

**[C]**E. Calabi,*An extension of E. Hopf’s maximum principle with an application to Riemannian geometry*, Duke Math. J.**25**(1958), 45–56. MR**0092069****[CLN]**Chow Bennett; Lu, Peng; Ni, Lei.*Hamilton's Ricci Flow*. Lectures in Contemporary Mathematics, Science Press, and Graduate Studies in Mathematics, Vol. 77, American Mathematical Society, Providence, RI, 2006.**[H]**Richard S. Hamilton,*A matrix Harnack estimate for the heat equation*, Comm. Anal. Geom.**1**(1993), no. 1, 113–126. MR**1230276**, 10.4310/CAG.1993.v1.n1.a6**[KL]**Karp, L. and Li, P.*The heat equation on complete riemannian manifolds*. Unpublished notes, 1982.**[KZ]**Kuang, S. and Zhang, Q. S.*A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow*. arXiv:math/0611298**[LY]**Peter Li and Shing-Tung Yau,*On the parabolic kernel of the Schrödinger operator*, Acta Math.**156**(1986), no. 3-4, 153–201. MR**834612**, 10.1007/BF02399203**[N]**Ni, Lei.*A note on Perelman's LYH inequality*.`arXiv:math.DG/0602337`**[N2]**Lei Ni,*The entropy formula for linear heat equation*, J. Geom. Anal.**14**(2004), no. 1, 87–100. MR**2030576**, 10.1007/BF02921867**[NT]**Lei Ni and Luen-Fai Tam,*Kähler-Ricci flow and the Poincaré-Lelong equation*, Comm. Anal. Geom.**12**(2004), no. 1-2, 111–141. MR**2074873****[P]**Perelman, Grisha.*The entropy formula for the Ricci flow and its Geometric Applications*.`arXiv:math.DG/0211159`**[S]**Wan-Xiong Shi,*Deforming the metric on complete Riemannian manifolds*, J. Differential Geom.**30**(1989), no. 1, 223–301. MR**1001277****[SZ]**Souplet, P. and Zhang, Q.S.*Sharp gradient estimate and Yau's Liouville theorem for the heat equation on non-compact manifolds*,`arXiv:math.DG/0502079`**[Z]**Qi S. Zhang,*Some gradient estimates for the heat equation on domains and for an equation by Perelman*, Int. Math. Res. Not. , posted on (2006), Art. ID 92314, 39. MR**2250008**, 10.1155/IMRN/2006/92314

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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.