TY - JOUR

T1 - Regularity of soap film-like surfaces spanning graphs in a Riemannian manifold

AU - Gulliver, Robert

AU - Park, Sung Ho

AU - Pyo, Juncheol

AU - Seo, Keomkyo

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Density

KW - Graph

KW - Soap film-like surface

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U2 - 10.4134/JKMS.2010.47.5.967

DO - 10.4134/JKMS.2010.47.5.967

M3 - Article

AN - SCOPUS:77958121213

SN - 0304-9914

VL - 47

SP - 967

EP - 983

JO - Journal of the Korean Mathematical Society

JF - Journal of the Korean Mathematical Society

IS - 5

ER -