Geometry & Topology

Existence of minimizing Willmore Klein bottles in Euclidean four-space

Patrick Breuning, Jonas Hirsch, and Elena Mäder-Baumdicker

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Abstract

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.

Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2485-2526.

Dates
Received: 6 April 2016
Revised: 8 August 2016
Accepted: 6 September 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1508437648

Digital Object Identifier
doi:10.2140/gt.2017.21.2485

Mathematical Reviews number (MathSciNet)
MR3654115

Zentralblatt MATH identifier
1365.53004

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C28: Twistor methods [See also 32L25] 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space

Keywords
Willmore surfaces Klein bottle

Citation

Breuning, Patrick; Hirsch, Jonas; Mäder-Baumdicker, Elena. Existence of minimizing Willmore Klein bottles in Euclidean four-space. Geom. Topol. 21 (2017), no. 4, 2485--2526. doi:10.2140/gt.2017.21.2485. https://projecteuclid.org/euclid.gt/1508437648


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