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

Abstract

Let α be a polygonal Jordan curve in \bm{R}³. We show that if α satisfies certain conditions, then the least-area Douglas-Radó disk in \bm{R}³ with boundary α is unique and is a smooth graph. As our conditions on α are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in \bm{R}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 \bm{R}³.