Existence of minimizing Willmore Klein bottles in Euclidean four-space

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 $f:K\to {ℝ}^n$ for $n\ge 4$ 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 $\mathcal{W}\left(f\right)\ge 8\pi$. 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 $8\pi$. 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.