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 $-κ^2$. Using the cone total curvature $TC(Γ)$ of a graph $Γ$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $Σ$ spanning a graph $Γ\subset M$ is less than or equal to $\frac{1}{2π}\{TC(Γ) - κ^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\times$}Γ)\}$. 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(Γ) < 3.649π+ κ^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}Γ}), \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,δ)$-minimizing set $Σ$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $π/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.