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

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 \(\bfR^3\).