Gradient estimates for positive solutions of the heat equation under geometric flow. (English) Zbl 1235.53070

Summary: We establish first- and second-order gradient estimates for positive solutions of the heat equations under general geometric flows. Our results generalize the recent work of S. Liu, who established similar results for the Ricci flow. Both results can also be considered as a generalization of P. Li, S. T. Yau, and J. Li’s gradient estimates under geometric flow setting. We also give an application to the mean curvature flow.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35K55 Nonlinear parabolic equations
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