# Minimizing 1/2-harmonic maps into spheres

## TL;DR

This paper advances the understanding of regularity and singularities of minimizing 1/2-harmonic maps into spheres, establishing smoothness in certain dimensions and classifying singularities in two dimensions.

## Contribution

It improves partial regularity results for 1/2-harmonic maps into spheres and classifies singularities in two dimensions, including uniqueness of certain homogeneous solutions.

## Key findings

- Minimizing 1/2-harmonic maps are smooth in 2D for target spheres of dimension ≥2.
- Singular sets have codimension at least 3 in higher dimensions.
- Unique non-trivial 0-homogeneous minimizers from the plane into the circle are rotations of x/|x|.

## 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 $m\geq 3$, 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 $\pm1$.