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 -