Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds

Abstract

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 Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville 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.

Hamilton Type Gradient Estimate for the Sub-Elliptic Operators

“…Assume the Ricci curvature is bounded below, i.e. Ricci ≥ −K for some constant K ≥ 0, Hamilton [14] obtained the following gradient estimate on compact Riemannian manifolds M: Later, B. L. Kotschwar [17] generalized this type of global elliptic gradient estimate on compact Riemannian manifolds to the case of noncompact manifolds; Souplet and Zhang [25] obtained local version of Hamilton type gradient estimate on complete noncompact Riemannian manifolds. Moreover, if the Ricci curvature is nonnegative, applying the doubling volume property and the lower (and upper) bounds of the heat kernel which are the consequences of Li-Yau inequality [20,21], we get the following estimate for the heat kernel:…”

Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds

“…For the purpose of obtaining the gradient estimates as in [14], we need to have the coefficient of w 2 be positive. Fortunately, we can do this by choosing a suitable β.…”

Harnack estimates for a nonlinear parabolic equation under Ricci flow

“…Furthermore, by the properties of Ψ and the assumption of the Ricci curvature, one has (cf., [12,15])…”