The Geometry of Genus-One Helicoids

Details The Geometry of Genus-One Helicoids The Geometry of Genus-One Helicoids

2007-07-16

arxiv.org/abs/0707.2393v2

Comment. Math. Helv. 84 (2009), no. 3, 547--569

Mathematics

Differential Geometry

22 pages. This updated version (Apr 17, 2009) contains a much simplified statement and proof of Lemma 3.2. This version will a

Scientific paper

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.

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