Vol. 243, No. 1, 2009

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Gradient estimates for solutions of the heat equation under Ricci flow

Shiping Liu

Vol. 243 (2009), No. 1, 165–180

We establish first order gradient estimates for positive solutions of the heat equations on complete noncompact or closed Riemannian manifolds under Ricci flows. These estimates improve Guenther’s results by weakening the curvature constraints. We also obtain a result for arbitrary solutions on closed manifolds under Ricci flows. As applications, we derive Harnack- type inequalities and second order gradient estimates for positive solutions of the heat equations under Ricci flow. The results in this paper can be considered as generalizing the estimates of Li–Yau and J. Y. Li to the Ricci flow setting.

gradient estimate, Ricci flow, heat equation, Harnack inequality
Mathematical Subject Classification 2000
Primary: 58J35, 53C44
Secondary: 35K55, 53C21
Received: 10 September 2008
Revised: 28 September 2008
Accepted: 20 May 2009
Published: 1 November 2009
Shiping Liu
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100190