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 Math. Z. 2006, 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 $frac{1}{2pi}{TC(Gamma) - kappa^{2}area(pmbox{$timeshspace*{-0.178cm}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 begin{eqnarray*} TC(Gamma) < 3.649pi + kappa^2 inf_{pin M} area({pmbox{$timeshspace*{-0.178cm}times$}Gamma}), end{eqnarray*} then the only possible singularities of a piecewise smooth $(mathbf{M},0,delta)$-minimizing set $Sigma$ is 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.