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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

Let K = P2 P2 be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles f : K n for n 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 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 W(f) 8π. 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 4 that have Euler normal number 4 or + 4 and Willmore energy 8π. The surfaces are distinct even when we allow conformal transformations of 4. As they are all minimizers in their regular homotopy class, they are Willmore surfaces.

Willmore surfaces, Klein bottle
Mathematical Subject Classification 2010
Primary: 53C42
Secondary: 53C28, 53A07
Received: 6 April 2016
Revised: 8 August 2016
Accepted: 6 September 2016
Published: 19 May 2017
Proposed: Tobias H. Colding
Seconded: Ian Agol, Leonid Polterovich
Patrick Breuning
Pastor-Felke-Str. 1
D-76131 Karlsruhe
Jonas Hirsch
Scuola Internazionale Superiore di Studi Avanzati
Via Bonomea, 265
34136 Trieste
Elena Mäder-Baumdicker
Institute for Analysis
Karlsruhe Institute of Technology
Englerstr. 2
D-76131 Karlsruhe