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Let $M$ be an $n$-dimensional complete simply connected Riemannian manifoldwith sectional curvature bounded above by a nonpositive constant $-\kappa^2$.Using the cone total curvature $TC(\Gamma)$ of a graph $\Gamma$ which wasintroduced by Gulliver and Yamada Math. Z. 2006, we prove that the density atany point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma \subsetM$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma) -\kappa^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\times$}\Gamma)\}$. From thisdensity estimate we obtain the regularity theorems for soap film-like surfacesspanning graphs with small total curvature. In particular, when $n=3$, thisdensity estimate implies that if \begin{eqnarray*} TC(\Gamma) < 3.649\pi +\kappa^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}\Gamma}),\end{eqnarray*} then the only possible singularities of a piecewise smooth$(\mathbf{M},0,\delta)$-minimizing set $\Sigma$ is the $Y$-singularity cone. Ina manifold with sectional curvature bounded above by $b^2$ and diameter boundedby $\pi/b$, we obtain similar results for any soap film-like surfaces spanninga graph with the corresponding bound on cone total curvature.

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