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

TL;DR

This paper proves that weakly stationary-harmonic multiple-valued functions in two dimensions are continuous inside their domain, extending regularity results related to Almgren's work on mass-minimizing currents.

Contribution

It introduces the class of weakly stationary-harmonic multiple-valued functions and establishes their interior continuity in two dimensions.

Findings

Weakly stationary-harmonic functions are continuous inside two-dimensional domains.

Extension of Almgren's regularity results to a broader class of functions.

Provides a new regularity result for multiple-valued functions in geometric measure theory.

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 $\Omega$.