Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

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 $ 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$.


References [Enhancements On Off] (What's this?)

  • [C] Calabi, E. An extension of E. Hopf's maximum principle with an application to Riemannian geometry. Duke Math. J. 25 (1957) 45-56.MR 0092069 (19:1056e)
  • [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] Hamilton, Richard S. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1 (1993), no.1, 113-126. MR 1230276 (94g:58215)
  • [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 Unpublished notes, 1982.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] Li, Peter and Yau, S.-T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153-201. MR 0834612 (87f:58156)
  • [N] Ni, Lei. A note on Perelman's LYH inequality. arXiv:math.DG/0602337
  • [N2] Ni, Lei. The entropy formula for linear heat equation. J. Geom Anal. 14 (2004), no. 1, 87-100.MR 2030576 (2004m:53118a)
  • [NT] Ni, Lei and Tam, L. F. Kähler-Ricci flow and the Poincarè-Lelong equation. Comm. Anal. Geom 12 (2004), no. 1-2, 111-141.MR 2074873 (2005f:53108)
  • [P] Perelman, Grisha. The entropy formula for the Ricci flow and its Geometric Applications. arXiv:math.DG/0211159
  • [S] Shi, Wan-Xiong Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989), no. 1, 223-301. MR 1001277 (90i:58202)
  • [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] 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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J35, 35K05

Retrieve articles in all journals with MSC (2000): 58J35, 35K05


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.

American Mathematical Society