Embedded Minimal Ends Asymptotic to the Helicoid

In [5], the genus one helicoid was constructed and strong evidence was given that it is embedded. The surface has infinite total curvature and contains two orthogonal, intersecting straight lines. Its single end has Weierstrass data modeled on that of the helicoid. It is conformally a punctured disk, on which both dh and dg/g have double poles at the puncture. Moreover, these forms have no residues. In this paper we study this type of end. Our main result is the following.

Let E be a complete, minimal annular end that is conformally a punctured disk. Suppose dg/g and dh each have a double pole at the puncture and that dh has no residue. If E contains a vertical ray and a horizontal ray, then a sub-end of E is embedded and is asymptotic to a helicoid.

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Published 1 January 2003