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

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related 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 Lionville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Lionville theorem for the heat equation does not hold even in R-n without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.