Minimizing 1/2-harmonic maps into spheres

Abstract

In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $m\ge 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle ${\mathbb{S}}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into ${\mathbb{S}}^1$ has a topological charge equal to $\pm 1$.