The Willmore energy of non-orientable surfaces

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.