A note on Willmore minimizing Klein bottles in Euclidean space

Hirsch, Jonas; Mäder-Baumdicker, Elena

doi:10.48550/arxiv.1606.04745

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2017

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“…We conclude that β n 2 ≤ 6π E 2 √ 2 3 ≈ 6.682π < 8π. There is some indication that τ3,1 is the actual minimizer among immersed Klein bottles in R 4 , compare the forthcoming paper [8].…”

Section: Introductionmentioning

confidence: 89%

Existence of minimizing Willmore Klein bottles in Euclidean four-space

Breuning¹,

Hirsch

Mäder-Baumdicker

2017

Geom. Topol.

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Abstract. 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 → R n , n ≥ 4, is attained by a smooth embedded Klein bottle. We know from [21,9] that there are three distinct regular homotopy classes of immersions f : K → 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 ) ≥ 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 R 4 that have Euler normal number −4 or +4 and Willmore energy 8π. 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.

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Section: Introductionmentioning

confidence: 89%