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

Let $\alpha$ be a polygonal Jordan curve in ${\text{bfR}}^3$. We show that if $\alpha$ satisfies certain conditions, then the least-area Douglas-Rad'{o} disk in ${\text{bfR}}^3$ 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 ${\text{bfR}}^3$ 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 ${\text{bfR}}^3$.