Existence and soap film regularity of solutions to Plateau’s problem

Abstract

AbstractPlateau’s problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in ${\mathbb{R}^{n}}$ in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension ${n-2}$ and ${\mathfrak{S}}$ the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of “size,” we prove: There exists an ${X_{0}}$ in ${\mathfrak{S}}$ with smallest size. Any such ${X_{0}}$ contains a “core” ${X_{0}^{*}\in\mathfrak{S}}$ with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of ${(n-1)}$-dimensional Hausdorff measure) a real analytic ${(n-1)}$-dimensional minimal submanifold. If ${n=3}$, then ${X_{0}^{*}}$ has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra.