Symmetry of embedded genus 1 helicoids

Abstract

In this article, we use the Lopez-Ros deformation to show that any embedded genus $1$ helicoid (or “genus-one helicoid”) must be symmetric with respect to rotation by $180°$ around a normal line. This partially answers a conjecture of Bobenko. We also show that this symmetry holds for an embedded genus $k$ helicoid $\Sigma$, provided that the underlying conformal structure of $\Sigma$ is hyperelliptic.