The Willmore conjecture was proved by F.\ Marques and A.\ Neves in 2012. Since then we

know that the Clifford torus has the lowest Willmore energy among all tori in $\mathbb R^3$.

I will present results concerning immersed Klein bottles in $\mathbb R^n$ with low Willmore

energy, $n\geq 4$. For Klein bottles immersed in $\mathbb R^4$ it is known that there are three

distinct regular homotopy classes each one containing an embedding. I will explain that

one of these classes contains the embedded Klein bottle with lowest Willmore energy among

all immersed Klein bottles. This minimizer has Willmore energy less than $7\pi$ and there

is a conjecture which surface it actually is. In the other two regular homotopy classes the

lowest possible value of the Willmore energy is $8 \pi$. There are infinitely many embedded

Klein bottles attaining this infimum. The talk is based on joint work with Jonas Hirsch and

Patrick Breuning.

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