Nonexistence of quasi-harmonic spheres with large energy

The absence 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 manifold. Assume the universal covering of (N, h) admits a nonnegative strictly convex function with polynomial growth. Then there is no non-constant quasi-harmonic sphere ${u:\mathbb{R}^n\rightarrow N}$ such that $$\lim_{r \rightarrow \infty}r^ne^{-\frac{r^2}{4}}\int \limits_{|x|\leq r}e^{-\frac{|x|^2}{4}}|\nabla u|^2{\text {d}}x\,=\,0.$$ This generalizes a result of the first author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus comparatively elementary.