Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in R ⁿ 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 ∑ spanning Γ. ∑ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature [InlineMediaObject not available: see fulltext.](Γ) for such graphs, or nets, Γ. We show that 2π times the area density of ∑ at any point is less than or equal to [InlineMediaObject not available: see fulltext.](Γ). For n=3, these density estimates imply, for example, that if [InlineMediaObject not available: see fulltext.](Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set ∑ are curves, along which three smooth sheets of ∑ meet with equal angles of 120^°.