  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.
