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 Q _Q (? ⁿ )-valued functions in the Sobolev space \(\mathcal{Y}_{2}(\varOmega, \mathbf{Q}_{Q} (\mathbb{R}^{n}))\) , where the domain Ω is open in ? ^m . In this article, we introduce the class of weakly stationary-harmonic Q _Q (? ⁿ )-valued functions. These functions are the critical points of Dirichlet’s integral under smooth domain-variations and range-variations. We prove that if Ω is a two-dimensional domain in ?² and \(f\in\mathcal{Y}_{2} (\varOmega,\mathbf{Q}_{Q}(\mathbb{R}^{n}) )\) is weakly stationary-harmonic, then f is continuous in the interior of the domain Ω.