The Geometry of Genus-One Helicoids
arXiv:0707.2393
Abstract
We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity.
22 pages. This updated version (Apr 17, 2009) contains a much simplified statement and proof of Lemma 3.2. This version will appear in Comm. Math. Helv