Nonexistence of quasi-harmonic sphere with large energy

Jiayu Li; Yunyan Yang

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arXiv · 1010.2407

Nonexistence of quasi-harmonic sphere with large energy

Jiayu Li·Yunyan Yang

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Abstract

Nonexistence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let $(N,h)$ be a complete noncompact Riemannian manifolds. Assume the universal covering of $(N,h)$ admits a nonnegative strictly convex function with polynomial growth. Then there is no quasi-harmonic spheres $u:\mathbb{R}^n\ra N$ such that $$\lim_{r\ra\infty}r^ne^{-\f{r^2}{4}}\int_{|x|\leq r}e^{-\f{|x|^2}{4}}|\nabla u|^2dx=0.$$ This generalizes a result of the first named author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus very simple.

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BibTeX RIS

Jiayu Li, Yunyan Yang. 2010-10-12. Nonexistence of quasi-harmonic sphere with large energy. https://arxiv.org/abs/1010.2407

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