Liouville theorems for self-similar solutions of heat flows

  • Meng Wang

    Zhejiang University, Hangzhou, China
  • Jiayu Li

    Chinese Academy of Sciences, Beijing, China


Let NN be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from (Rm,ex2/2(m2)ds02)({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2) to NN (m3m\geq 3) with finite energy ([LnW]). Here ds02ds_0^2 is the Euclidean metric in Rm{\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 NN. 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