The Geometry of Genus-One Helicoids

TL;DR

This paper characterizes genus-one minimal surfaces asymptotic to helicoids, proving they intersect the helicoid only along specified lines and establishing conditions for asymptotic behavior at infinity.

Contribution

It provides new geometric insights into genus-one minimal surfaces related to helicoids, including intersection properties and asymptotic conditions.

Findings

Properly embedded genus-one minimal surfaces intersect the helicoid only along two lines.

Two straight lines divide the surface into two components on either side of the helicoid.

A simple condition ensures minimal surfaces with finite genus are asymptotic to a helicoid.

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.