A Multiple-Valued Plateau Problem

The existence of Dirichlet energy minimizing multiple-valued functions for given Dirichlet boundary data has been known since pioneering work [1] of F. J. Almgren. Here, we prove a multiple-valued analogue of the classical Plateau problem on the existence of area-minimizing mappings of the disk. Specifically, we find, for positive integers K, l(1), ..., l(k), Q = l(1)+, ..., +l(k), and for any collection of K disjoint smoothly embedded curves, Gamma(1), ..., Gamma(k), Q-valued Plateau boundary data with wrapping numbers l(i) about Gamma(i), which extends to a Dirichlet minimizing Q-valued function with minimal Dirichlet energy among all possible monotone parameterizations of the boundary curves. Conformality holds for such minimizers under a nondegeneracy condition analogous to the Douglas condition for minimizers from planar domains. Finally, we analyze two particular cases where, in contrast to single-valued Douglas solutions in R-3, we have a class of examples for which our multiple-valued Plateau solution have branch points. Second, we give examples of a degenerate behavior, illustrating the weakness of a possible multiple-valued maximum principle and providing motivation for defining our analogous Douglas condition.