Liouville theorems for self-similar solutions of heat flows

Abstract

Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from $({\mathbf{R}}^m,e^{-|x{|}^2/2(m-2)}d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $d{s}_0^2$ is the Euclidean metric in ${\mathbf{R}}^m$. It arises from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.