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

TL;DR

This paper establishes density bounds and regularity results for soap film-like surfaces spanning graphs in Riemannian manifolds with nonpositive curvature, identifying conditions under which singularities are limited to Y-shaped cones.

Contribution

It introduces a density estimate based on cone total curvature that leads to new regularity theorems for soap film-like surfaces in curved manifolds.

Findings

Density at points is bounded by cone total curvature and area terms.

Regularity theorems hold for graphs with small total curvature.

Singularities are limited to Y-cones under certain curvature conditions.

Abstract

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-\kappa^2$. Using the cone total curvature $TC(\Gamma)$ of a graph $\Gamma$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma \subset M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma) - \kappa^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\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 \begin{eqnarray*} TC(\Gamma) < 3.649\pi + \kappa^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}\Gamma}), \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,\delta)$-minimizing set $\Sigma$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.