Nonexistence of quasi-harmonic sphere with large energy

TL;DR

This paper proves the nonexistence of certain quasi-harmonic spheres under specific geometric conditions, which is important for understanding the behavior of harmonic map heat flows.

Contribution

It generalizes previous nonexistence results for quasi-harmonic spheres to broader manifolds with polynomial growth conditions.

Findings

No quasi-harmonic spheres exist under given conditions.

The proof uses a simple Moser iteration method.

Extends earlier results to more general noncompact manifolds.

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.