#### Volume 21, issue 4 (2017)

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Existence of minimizing Willmore Klein bottles in Euclidean four-space

### Patrick Breuning, Jonas Hirsch and Elena Mäder-Baumdicker

Geometry & Topology 21 (2017) 2485–2526
##### Abstract

Let $K=ℝ{P}^{2}♯ℝ{P}^{2}$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles $f:K\to {ℝ}^{n}$ for $n\ge 4$ is attained by a smooth embedded Klein bottle. We know from work of M W Hirsch and W S Massey that there are three distinct regular homotopy classes of immersions $f:K\to {ℝ}^{4}$, 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 $\mathsc{W}\left(f\right)\ge 8\pi$. We give a classification of the minimizers of these two regular homotopy classes. In particular, we prove the existence of infinitely many distinct embedded Klein bottles in ${ℝ}^{4}$ that have Euler normal number $-4$ or $+4$ and Willmore energy $8\pi$. The surfaces are distinct even when we allow conformal transformations of ${ℝ}^{4}$. As they are all minimizers in their regular homotopy class, they are Willmore surfaces.

##### Keywords
Willmore surfaces, Klein bottle
##### Mathematical Subject Classification 2010
Primary: 53C42
Secondary: 53C28, 53A07