Let be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from to () with finite energy ([LnW]). Here is the Euclidean metric in . 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 . We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.
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Meng Wang, Jiayu Li, Liouville theorems for self-similar solutions of heat flows. J. Eur. Math. Soc. 11 (2009), no. 1, pp. 207–221DOI 10.4171/JEMS/147