# Liouville theorems for self-similar solutions of heat flows

### Meng Wang

Zhejiang University, Hangzhou, China### Jiayu Li

Chinese Academy of Sciences, Beijing, China

## Abstract

Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from $({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2)$ to $N$ ($m\geq 3$) with finite energy ([LnW]). Here $ds_0^2$ is the Euclidean metric in ${\bf R}^m$. 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.

## Cite this article

Meng Wang, Jiayu Li, Liouville theorems for self-similar solutions of heat flows. J. Eur. Math. Soc. 11 (2009), no. 1, pp. 207–221

DOI 10.4171/JEMS/147