A gap theorem for α-harmonic maps between two-spheres

Abstract

We consider approximations introduced by Sacks and Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue our previous analysis (J. Differential Geom. 116:2 (2020), 321–348) on limits of $\alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity of Li and Zhu (Ann. Inst. H. Poincaré Anal. Non Linéaire 36:1 (2019), 103–118), we obtain an optimal gap theorem for the $\alpha$-harmonic maps of degree $-1,0$ or $1$.