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 $-{\kappa}^2$. 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 $\Gamma\;\subset\;M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}$. 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 $TC(\Gamma)$ < $3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})$, 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.