Regularity of soap film-like surfaces spanning graphs in a Riemannian manifold

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant -κ². Using the cone total curvature TC(Γ) of a graph Γ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface ∑ spanning a graph Γ ⊂ M is less than or equal to 1/2π {TC(Γ}) - κ²Area(p×Γ)}. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3, this density estimate implies that if TC(Γ) < 3.649 π + κ² inf p ⊂ M Area(p×Γ), then the only possible singularities of a piecewise smooth (M, 0, δ)-minimizing set ∑ are the Y -singularity cone. In a manifold with sectional curvature bounded above by b² and diameter bounded by π/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.