Existence of minimizing Willmore Klein bottles in Euclidean four-space

Breuning, Patrick; Hirsch, Jonas; Mäder-Baumdicker, Elena

doi:10.2140/gt.2017.21.2485

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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

DOI: 10.2140/gt.2017.21.2485

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 $n \geq 4$ 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 $W (f) \geq 8 π$ . 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 $8 π$ . 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

References

Publication

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

Authors

Patrick Breuning
Pastor-Felke-Str. 1 D-76131 Karlsruhe Germany

Jonas Hirsch
Scuola Internazionale Superiore di Studi Avanzati Via Bonomea, 265 34136 Trieste Italy

Elena Mäder-Baumdicker
Institute for Analysis Karlsruhe Institute of Technology Englerstr. 2 D-76131 Karlsruhe Germany

Volume 21, issue 4 (2017)