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

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

2008-04-26

arxiv.org/abs/0804.4208v1

J. Math. Soc. Japan 52(1) (2000), 25-40

Mathematics

Differential Geometry

Scientific paper

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 $\bfR^3$.

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