Nonexistence of quasi-harmonic sphere with large energy

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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:Rⁿ N such that ∞rⁿe-r²4∫|ₓ|≤ ᵣe-|x|²4|∇ 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.