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

- Journal title : Journal of the Korean Mathematical Society
- Volume 47, Issue 5, 2010, pp.967-983
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2010.47.5.967

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;

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 . Using the cone total curvature TC() of a graph 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 is less than or equal to . 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 < , then the only possible singularities of a piecewise smooth (M, 0, )-minimizing set are the Y-singularity cone. In a manifold with sectional curvature bounded above by and diameter bounded by /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

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