Symmetry of Embedded Genus-One Helicoids

TL;DR

This paper proves that embedded genus-one helicoids are symmetric under 180-degree rotation, using Lopez-Ros deformation, and extends this symmetry result to genus-k helicoids with hyperelliptic conformal structure.

Contribution

It demonstrates the symmetry of embedded genus-one helicoids and extends the result to genus-k helicoids with hyperelliptic conformal structure, partially confirming a conjecture.

Findings

Embedded genus-one helicoids are symmetric under 180-degree rotation.

Symmetry also holds for genus-k helicoids with hyperelliptic conformal structure.

The Lopez-Ros deformation is used to establish these symmetries.

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.