Existence of minimizing Willmore Klein bottles in Euclidean four-space

TL;DR

This paper proves the existence of minimal Willmore energy Klein bottles in four-dimensional Euclidean space, classifies their types, and demonstrates infinitely many such minimizers with specific normal Euler numbers.

Contribution

It establishes the existence and classification of minimizers of Willmore energy among Klein bottles in four-space, including infinitely many distinct embedded minimizers.

Findings

Existence of smooth embedded Klein bottles minimizing Willmore energy in 4D.

Classification of Klein bottles into three regular homotopy classes.

Existence of infinitely many minimizers with specific Euler normal numbers.

Abstract

Let $K$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles in Euclidean $n$-space is attained by a smooth embedded Klein bottle, where $n\geq 4$. There are three distinct regular homotopy classes of immersed Klein bottles in the Euclidean four-space each one containing an embedding. One is characterized by the property that it contains the minimizer just mentioned. For the other two regular homotopy classes we show that the Willmore energy is bounded from below by $8\pi$. We give a classification of the minimizers of these two classes. In particular, we prove the existence of infinitely many distinct embedded Klein bottles in Euclidean four-space that have Euler normal number $-4$ or $+4$ and Willmore energy $8\pi$. The surfaces are distinct even when we allow conformal transformations of the ambient space. As they are all minimizers in their homotopy class they are Willmore surfaces.