Geometry & Topology
- Geom. Topol.
- Volume 21, Number 4 (2017), 2485-2526.
Existence of minimizing Willmore Klein bottles in Euclidean four-space
Let be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles for 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 , 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 . 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 that have Euler normal number or and Willmore energy . The surfaces are distinct even when we allow conformal transformations of . As they are all minimizers in their regular homotopy class, they are Willmore surfaces.
Geom. Topol., Volume 21, Number 4 (2017), 2485-2526.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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