Willmore Energy Estimates for Klein Bottles

In this note, we provide an estimate for the Willmore energy of Klein bottles in ${\mathbb{R}}^N$, building on the ideas of Li-Yau and Montiel-Ros for $2$-tori. In particular, we prove that the Willmore energy $\mathcal{W}(\phi )>6\pi$ for any conformal branched immersion $\phi :{\mathbb{K}}_b^2\to {\mathbb{R}}^N$, where ${\mathbb{K}}_b^2$ is a flat Klein bottle with $0.350\lesssim b\lesssim 0.755$. This confirms partially a conjecture of Kusner. Moreover, for a flat immersion $\psi :{\mathbb{K}}_b^2\to {\mathbb{R}}^N$, we derive a lower bound of $\mathcal{W}(\psi )$, which confirms a conjecture of Hirsch-Mäder-Baumdicker for flat Klein bottles in ${\mathbb{R}}^N$. We also construct a smooth family of flat Klein bottles in ${\mathbb{R}}^5$, with Willmore energy between $6.912\pi$ and $8\pi$.