Journal of the
Korean Mathematical Society JKMS

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 ${\frac{1}{2\pi}\{TC(\Gamma) - \kappa^{2}{\rm Area}(p\mbox{$\times\!\!\!\!\!\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) < 3.649\pi + \kappa^2 \displaystyle{\inf_{p\in M} {\rm Area}({p\mbox{$\times\!\!\!\!\!\times$}\Gamma})}, \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,\delta)$-minimizing set $\Sigma$ are the $Y$-singularity cone. and the $T$-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.

Keywords: soap film-like surface, graph, density

MSC numbers: 58E35, 49Q20