Classical solutions to the soap film capillarity problem for plane boundaries

TL;DR

This paper establishes existence, uniqueness, and geometric properties of classical soap film solutions within tubular neighborhoods of smooth curves, demonstrating their structure as constant mean curvature hypersurfaces with orthogonal boundary contact.

Contribution

It provides the first rigorous proof of classical minimizers' existence and uniqueness for soap films with tubular boundary conditions, detailing their geometric structure and foliation properties.

Findings

Existence and uniqueness of classical minimizers for soap film capillarity problems.

The minimizers' boundaries are symmetric smooth graphs with constant mean curvature.

Solutions form a foliation of space by constant mean curvature hypersurfaces.

Abstract

We study the soap film capillarity problem, in which soap films are modeled as sets of least perimeter among those having prescribed (small) volume and satisfying a topological spanning condition. When the given boundary is the closed tubular neighborhood in $\mathbb{R}^3$ of a smooth Jordan curve (or, more generally, the closed tubular neighborhood in $\mathbb{R}^d$ of a smooth embedding of $\mathbb{S}^{d-2}$ in a hyperplane), we prove existence and uniqueness of classical minimizers, for which the collapsing phenomenon does not occur. We show that the boundary of the unique minimizer is the union of two symmetric smooth normal graphs over a portion of the plane; the graphs have positive constant mean curvature bounded linearly in terms of the volume parameter, and meet the boundary of the tubular neighbourhood orthogonally. Moreover, we prove uniform bounds on the sectional curvatures in order to show that the boundaries of solutions corresponding to varying volumes are ordered monotonically and produce a foliation of space by constant mean curvature hypersurfaces.