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On embeddedness of area-minimizing disks, and an application to constructing complete minimal surfaces. (English) Zbl 0946.53005

Let \(\alpha\) be a polygonal Jordan curve in \(\mathbb{R}^3\). We show that if \(\alpha\) satisfies certain conditions, then the least-area Douglas-Radó disk in \(\mathbb{R}^3\) 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 \(\mathbb{R}^3\) 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 \(\mathbb{R}^3\).
Reviewer: W.Rossman (Kobe)

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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