The geometry of genus-one helicoids

  • David Hoffman

    Stanford University, United States
  • Brian White

    Stanford University, United States


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.

Cite this article

David Hoffman, Brian White, The geometry of genus-one helicoids. Comment. Math. Helv. 84 (2009), no. 3, pp. 547–569

DOI 10.4171/CMH/172