Mathematics

The Willmore energy of an immersed surface $f: \Sigma \to \mathbb R^n$, $n \ge 3$, is defined to be
\[ \mathcal W(f):= \frac{1}{4} \int_{\Sigma} \vert H \vert ^2 \, d\mu_g. \]
Willmore proposed in the 1960's that an immersion that minimizes the energy over all possible immersions has presumably the optimal shape in the sense of bending energy.
So it is almost natural to ask what is the optimal shape of a Klein bottle.
In a first part I want to give glimpse on a compactness result of $W^{2,2}$-conformal immersions of a closed Riemann surface into $\mathbb R^n$ that has been proven by Kuwert and Li. Afterwards I want to discuss how it can be used to find an "optimal" Klein bottle in Euclidian four space and discuss why it is more difficult in Euclidean three space.
If time permits I will present a candidate for the minimizer.