Area Density and Regularity for Soap Film-like Surfaces Spanning Graphs

For a given boundary Γ in R consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature Ctot(Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to Ctot(Γ). For n = 3, these density estimates imply, for example, that if Ctot(Γ) ≤ 3.649π, then the only possible singularities of a piecewise smooth (M, 0, δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120◦.