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

Let alpha be a polygonal Jordan curve in R-3. We show that if alpha satisfies certain conditions, then the least-area Douglas-Rado disk in R-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 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 R-3.