Minimizing 1/2harmonic 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 $(m1)$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 $\pm1$.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.05790
 Bibcode:
 2019arXiv190105790M
 Keywords:

 Mathematics  Analysis of PDEs