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

TL;DR

This paper extends a 1998 result by proving the existence of infinitely many homotopy classes of fractional harmonic maps from $ ext{S}^3$ to $ ext{S}^2$ with minimizing $W^{s,3/s}$-harmonic maps for $s$ in (0,1).

Contribution

It generalizes the existence of minimizing harmonic maps to fractional harmonic maps, revealing infinitely many homotopy classes with such minimizers.

Findings

Infinitely many homotopy classes admit minimizing fractional harmonic maps.

Extension of classical harmonic map results to fractional Sobolev spaces.

Existence results hold for all $s$ in (0,1).

Abstract

In 1998 T. Rivi\`{e}re 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.