Symmetry of Embedded Genus-One Helicoids

Abstract

In this note, we use the Lopez-Ros deformation introduced in [9] to show that any embedded genus-one helicoid must be symmetric with respect to rotation by 180 degrees around a normal line. This partially answers a conjecture of Bobenko from [3]. We also show this symmetry holds for an embedded genus-k helicoid $\Sigma$, provided the underlying conformal structure of $\Sigma$ is hyperelliptic.