Existence of infinitely many homotopy classes from ${\mathbb{S}}^3$ to ${\mathbb{S}}^2$ having a minimimzing $W^{s,\frac{3}{s}}$-harmonic map

Abstract

In 1998 T. Rivière proved that there exist infinitely many homotopy classes of ${\pi }_3({\mathbb{S}}^2)$ having a minimizing 3-harmonic map. This result is especially surprising taking into account that in ${\pi }_3({\mathbb{S}}^3)$ there are only three homotopy classes (corresponding to the degrees $\{-1,0,1\}$) in which a minimizer exists. We extend this theorem in the framework of fractional harmonic maps and prove that for $s\in (0,1)$ there exist infinitely many homotopy classes of ${\pi }_3({\mathbb{S}}^2)$ in which there is a minimizing $W^{s,\frac{3}{s}}$-harmonic map.