A Gap Theorem for alpha-harmonic maps between two-spheres

In this paper we consider approximations `a la Sacks-Uhlenbeck of the harmonic energy for maps from S2 into S2. We continue the analysis in [6] about limits of α-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the α-harmonic maps of degree −1, 0 or 1.

We consider approximations introduced by Sacks and Uhlenbeck of the harmonic energy for maps from S 2 into S2. We continue our previous analysis (J. Differential Geom. 116:2 (2020), 321-348) on limits of a-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 a-harmonic maps of degree -1,0 or 1.