Regularity of soap film-like surfaces spanning graphs in a Riemannian manifold

Robert Gulliver, Sung Ho Park, Juncheol Pyo, Keomkyo Seo

Research output: Contribution to journalArticlepeer-review

Abstract

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant -κ2. 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}) - κ2Area(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 + κ2 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 b2 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 languageEnglish (US)
Pages (from-to)967-983
Number of pages17
JournalJournal of the Korean Mathematical Society
Volume47
Issue number5
DOIs
StatePublished - 2010

Keywords

  • Density
  • Graph
  • Soap film-like surface

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