Abstract

In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of Millot and Sire (Arch Ration Mech Anal 215:125–210, 2015), Moser( J Geom Anal 21:588–598, 2011) in the case where the target manifold is the \((m-1)\)-dimensional sphere. For \(m\geqslant 3\), 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 map from a 2d domain into \({\mathbb {S}}^1\) has a topological charge equal to \(\pm 1\).

中文翻译：

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最小化1/2谐波映射到球体中

摘要

在本文中，我们改进了部分正则性理论以最小化Millot和Sire的1/2调和图（Arch Ration Mech Anal 215：125–210，2015），Moser（J Geom Anal 21：588-598，2011）目标流形是\（（m-1）\）维球体的情况。对于\（m \ geqslant 3 \），我们表明最小化1/2谐谱在维度2上是平滑的，并且在更高维度上具有至少3个奇异的余维集。对于\（m = 2 \），我们证明，直到正交变换，x / | x | 是从平面到圆\（{\ mathbb {S}} ^ 1 \）的唯一的非平凡的0均匀最小化1/2谐波映射。因此，从2d域到\（{\ mathbb {S}} ^ 1 \）的最小1/2谐谱的每个点奇点具有等于\（\ pm 1 \）的拓扑电荷。