REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD

Abstract

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant -kappa(2). Using the cone total curvature TC(Gamma) of a graph Gamma which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface Sigma spanning a graph Gamma subset of M is less than or equal to 1/2 pi{TC(Gamma) - kappa(2)Area(p(sic)Gamma)}. 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(Gamma) < 3.649 pi + kappa(2) inf(p is an element of M) Area(p(sic)Gamma), then the only possible singularities of a piecewise smooth (M, 0, delta)-minimizing set Sigma are the Y-singularity cone. In a manifold with sectional curvature bounded above by b(2) and diameter bounded by pi/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.