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

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 $$\mathcal{Y}${2}(\Omega, $\mathbf{Q}${Q} ($\mathbb{R}$^{n}))$,wherethedomain$\Omega$isopenin$$\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 $\mathrm{\Omega }$ is a two-dimensional domain in ${\mathbb{R}}^2$ and $f\in {\mathcal{Y}}_2(\mathrm{\Omega },{\mathbf{Q}}_Q({\mathbb{R}}^n))$ is weakly stationary-harmonic, then $f$ is continuous in the interior of the domain $\mathrm{\Omega }$.