## 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 [8], we prove that the density at any point of a soap film-like surface ∑ spanning a graph Γ ⊂ M is less than or equal to 1/2π {TC(Γ}) - κ^{2}Area(p×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 TC(Γ) < 3.649 π + κ^{2} inf p ⊂ M Area(p×Γ), then the only possible singularities of a piecewise smooth (M, 0, δ)-minimizing set ∑ are 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.

Original language | English (US) |
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Pages (from-to) | 967-983 |

Number of pages | 17 |

Journal | Journal of the Korean Mathematical Society |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - 2010 |

## Keywords

- Density
- Graph
- Soap film-like surface