Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds

Details Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds

2005-02-03

arxiv.org/abs/math/0502079v1

Mathematics

Differential Geometry

Scientific paper

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive eternal solutions of the heat equation, under certain growth condition. Surprisingly, this Liouville theorem for the heat equation does not hold even in ${\bf R}^n$ without such a condition. We also prove a sharpened long time gradient estimate for the log of heat kernel on noncompact manifolds. This has been an open problem in view of the well known estimates in the compact, short time case.

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