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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.

Gradient estimates for solutions of the heat equation under Ricci flow