Existence of minimizing Willmore Klein bottles in Euclidean four-space

Breuning, Patrick¹ ; Hirsch, Jonas² ; Mäder-Baumdicker, Elena³

¹ Pastor-Felke-Str. 1, D-76131 Karlsruhe, Germany
² Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea, 265, 34136 Trieste, Italy
³ Institute for Analysis, Karlsruhe Institute of Technology, Englerstr. 2, D-76131 Karlsruhe, Germany

Geometry & topology, Tome 21 (2017) no. 4, pp. 2485-2526.

Résumé

Let $K=ℝP^2♯ℝP^2$ 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 $\mathcal{W}\left(f\right)\ge 8\pi$. 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.

Détail

DOI : 10.2140/gt.2017.21.2485

Classification : 53C42, 53C28, 53A07
Keywords: Willmore surfaces, Klein bottle

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