A Multiple-Valued Plateau Problem

Original · EN

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for K ∈ N, k₁,...,kₖ∈ N with sum Q and any collection of K disjoint Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary data with these multiplicities which extends to a Dirichlet minimizing Q-valued function with minimal Dirichlet energy among all possible monotone parameterizations of the boundary curves. Under a condition analogous to the Douglas condition for minimizers from planar domains, conformality of the minimizer follows from topological methods and some complex analysis. Finally, we analyze two particular cases: in contrast to single-valued Douglas solutions, we first give a class of examples for which our multiple-valued Plateau solution has branch points. Second, we give examples of a degenerate behavior, illustrating the weakness of the multiple-valued maximum principle and provide motivation for our analogous Douglas condition.