Inserted: 9 apr 2019
Last Updated: 9 apr 2019
In this paper we consider approximations à la Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in 6 about limits of $\alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity in 7, we obtain an optimal gap theorem for the $\alpha$-harmonic maps of degree $-1, 0$ or $1$.