Plateau's Problem via covering spaces

Abstract

In 1995, Brakke proposed a formulation of the Plateau problem for a given boundary $\Gamma$ that allows for triple junctions and tetrahedral singularities. Let $\Gamma$ be a smooth closed curve and let $G=\pi_1(\mathbb{R}^3\setminus \Gamma)$. For each proper finite index subgroup $N$ of $G$, Brakke constructs a $(\mathrm{\mathbf{M}}, 0, \infty)$-minimal surface $\Sigma_N$, which is obtained as the projection of the boundary of a perimeter-minimising fundamental domain in the covering space associated to $N$. We advance the theory in two ways. Firstly, we extend Brakke's construction to include all normal subgroups $N\triangleleft G$, i.e. possibly with infinite index. Secondly, we prove a compactness result which implies that there exists a proper normal subgroup $N_0\triangleleft G$ such that \[ \mathrm{Area}(\Sigma_{N_0})=\inf_{\{N\triangleleft G, N \neq G\}}\ \mathrm{Area}(\Sigma_N). \] A similar result holds when $\Gamma$ has many connected components. Furthermore, we study the spanning and minimising properties of the $\Sigma_N$s and $\Sigma_{N_0}$.