Interior continuity of two-dimensional weakly stationary-harmonic multiple-valued functions

Abstract

In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing ${\mathbf{Q}}_Q({\mathbb{R}}^n)$-valued functions in the Sobolev space ${\mathcal{Y}}_2(\Omega ,{\mathbf{Q}}_Q({\mathbb{R}}^n))$, where the domain $\Omega$ is open in ${\mathbb{R}}^m$. In this article, we introduce the class of weakly stationary-harmonic ${\mathbf{Q}}_Q({\mathbb{R}}^n)$-valued functions. These functions are the critical points of Dirichlet integral under smooth domain-variations and range-variations. We prove that if $\Omega$ is a two-dimensional domain in ${\mathbb{R}}^2$ and $f\in {\mathcal{Y}}_2(\Omega ,{\mathbf{Q}}_Q({\mathbb{R}}^n))$ is weakly stationary-harmonic, then $f$ is continuous in the interior of the domain $\Omega$.