A note on Willmore minimizing Klein bottles in Euclidean space

TL;DR

This paper proves that Lawson's bipolar surface is the unique Willmore minimizer among immersed Klein bottles in its conformal class and conjectures its uniqueness in higher-dimensional Euclidean spaces.

Contribution

It establishes the uniqueness of Lawson's bipolar surface as a Willmore minimizer in its conformal class and proposes a conjecture for higher dimensions.

Findings

Lawson's bipolar surface is the unique minimizer in its conformal class.

Conjecture: it is also the unique minimizer in all higher-dimensional Euclidean spaces.

Provides a proof of uniqueness after stereographic projection.

Abstract

We show that Lawson's bipolar surface $\tilde\tau_{3,1}$ is after stereographic projection the unique minimizer among immersed Klein bottles in its conformal class. We conjecture that it actually is the unique minimizer among immersed Klein bottles into $\mathbb{R}^n$, $n\geq 4$, whose existence the authors and P. Breuning proved in a previous paper.