A Multiple-Valued Plateau Problem

TL;DR

This paper extends the classical Plateau problem to multiple-valued functions, proving existence of area-minimizing solutions with prescribed boundary data and analyzing their properties, including branch points and degeneracies.

Contribution

It introduces a multiple-valued analogue of the Plateau problem, establishing existence, minimality, and conformality conditions for these solutions, and explores their unique features.

Findings

Existence of Dirichlet minimizing multiple-valued functions with prescribed boundary data.

Examples of solutions with branch points, unlike single-valued cases.

Illustration of degenerate behaviors and limitations of the maximum principle.

Abstract

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 $K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$ 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.