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

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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 ≥-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 ℝn.

Original languageEnglish (US)
Pages (from-to)3013-3019
Number of pages7
JournalProceedings of the American Mathematical Society
Volume135
Issue number9
DOIs
StatePublished - Sep 1 2007
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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