### 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 ℝ^{n}.

Original language | English (US) |
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Pages (from-to) | 3013-3019 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 135 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2007 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Hamilton's gradient estimate for the heat kernel on complete manifolds.** / Kotschwar, Brett.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 135, no. 9, pp. 3013-3019. https://doi.org/10.1090/S0002-9939-07-08837-5

}

TY - JOUR

T1 - Hamilton's gradient estimate for the heat kernel on complete manifolds

AU - Kotschwar, Brett

PY - 2007/9

Y1 - 2007/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77950637548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950637548&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-07-08837-5

DO - 10.1090/S0002-9939-07-08837-5

M3 - Article

VL - 135

SP - 3013

EP - 3019

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 9

ER -