Symmetry of embedded genus 1 helicoids

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 Σ , provided that the underlying conformal structure of Σ is hyperelliptic.