REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD

Title & Authors
REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD
Gulliver, Robert; Park, Sung-Ho; Pyo, Jun-Cheol; Seo, Keom-Kyo;

Abstract
Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $\small{-{\kappa}^2}$. Using the cone total curvature TC($\small{\Gamma}$) of a graph $\small{\Gamma}$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $\small{\Sigma}$ spanning a graph $\small{\Gamma\;\subset\;M}$ is less than or equal to $\small{\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}}$. 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 $\small{TC(\Gamma)}$ < $\small{3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})}$, then the only possible singularities of a piecewise smooth (M, 0, $\small{\delta}$)-minimizing set $\small{\Sigma}$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $\small{b^2}$ and diameter bounded by $\small{\pi}$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.
Keywords
soap film-like surface;graph;density;
Language
English
Cited by
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