A note on the nonexistence of quasi-harmonic spheres

Abstract

In this paper we study the properties of quasi-harmonic spheres from $\R^m, m>2$. We show that if the universal covering $\tilde N$ of $N$ admits a nonnegative strictly convex function $\rho$ with the exponential growth condition $\rho(y)\leq C\exp\left(\frac14\tilde d(y)^{2/m}\right)$ where $\tilde d(y)$ is the distance function on $\tilde N$, then $N$ does not admit a quasi-harmonic sphere, which generalize Li-Zhu's result \cite{Li2010non}. We also show that if $u$ is a quasi-harmonic sphere, then the property that $u$ is of finite energy ($\int_{\R^m}e(u)e^{-\abs{x}^2/4}\dif x<\infty$) is equivalent to the property that $u$ satisfies the large energy condition ($\lim_{R\to\infty}R^{m}e^{-R^2/4}\int_{B_R(0)}e(u)e^{-\abs{x}^2/4}\diff x=0$).