Genus-One Helicoids from a Variational Point of View

Abstract

We prove by variational means the existence of a complete, properly embedded, genus-one minimal surface in R^3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid's axis, but that have infinitely many handles arranged periodically along the axis. Finally, we prove some new properties of such helicoid-like surfaces.

36 pages, 5 figures. Revised version: typos corrected, references added, proof of Thm 6.1 made more self-contained, several paragraphs added to the proof of Theorem 6.2