Existence and Soap Film Regularity of Solutions to Plateau's Problem

Plateau's soap film problem is to find a surface of least area spanning a given boundary. We begin with a compact orientable $(n-2)$-dimensional submanifold $M$ of $\R^n$. If $M$ is connected, we say a compact set $X$ "spans" $M$ if $X$ intersects every Jordan curve whose linking number with $M$ is 1. Picture a soap film that spans a loop of wire. Using $(n-1)$-dimensional Hausdorff spherical measure as the measure of the size of a compact set $X$ in $\R^n$, we prove there exists a smallest compact set $X_0$ that spans $M$. We also show that $X_0$ is almost everywhere a real analytic $(n-1)$-dimensional minimal submanifold and if $n = 3$, then $X_0$ has the structure of a soap film as predicted by Plateau. We provide more details about the minimizer $X_0$. Primarily, $X_0$ is the support of a current $S_0$ and $M$ is the support of the algebraic boundary of $S_0$. We also discuss the more general case where $M$ has codimension $> 2$.