Continuity of two-dimensional weakly stationary-harmonic multiple-valued functions

In this talk, I would like to introduce the class of weakly stationary-harmonic multiple-valued functions in the Sobolev space of multiple-valued functions. This class of functions are defined as the critical points of Dirichlet integral under smooth domain-variations and range-variations.

I would show the interior continuity for this class of functions as the domain-dimension is two.