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Abstract
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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.
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Keywords
gradient estimate, Ricci flow, heat
equation, Harnack inequality
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Mathematical Subject Classification
Primary: 58J35, 53C44
Secondary: 35K55, 53C21
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Milestones
Received: 10 September 2008
Revised: 28 September 2008
Accepted: 20 May 2009
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