On embeddedness of area-minimizing disks, and an application to constructing complete minimal surfaces

TL;DR

This paper establishes new conditions under which area-minimizing disks bounded by certain polygonal curves in three-dimensional space are unique and embedded, expanding the class of known minimal surfaces and enhancing the conjugate surface construction method.

Contribution

It introduces novel conditions for the embeddedness of least-area disks bounded by polygonal Jordan curves, broadening the applicability of minimal surface construction techniques.

Findings

Unique, smooth, embedded minimal disks for new classes of boundary curves.

Extended applicability of conjugate surface method to more minimal surfaces.

Broadened understanding of embeddedness conditions for area-minimizing disks.

Abstract

Let $\alpha$ be a polygonal Jordan curve in $\bfR^3$. We show that if $\alpha$ satisfies certain conditions, then the least-area Douglas-Rad\'{o} disk in $\bfR^3$ with boundary $\alpha$ is unique and is a smooth graph. As our conditions on $\alpha$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces in $\bfR^3$.