On the Willmore energy of M\"{o}bius bands

Abstract

We show that among M\"obius bands in $\mathbb{S}^3$ bounded by a great circle, the minimal Willmore energy is realized by an embedded minimal M\"obius band with Morse index two. To prove this, we introduce a $2$-parameter ``canonical family"associated to any non-orientable surface in $\mathbb{S}^3$ with boundary a great circle and apply a min-max argument. The canonical family detects the Euler number of the surface and is inspired by the $5$-parameter family discovered by F.C. Marques and A. Neves detecting the genus of an orientable surface in $\mathbb{S}^3$. For $\mathbb{Z}_2$-invariant Klein bottles, this reduces R. Kusner's 1989 conjecture that $\tau_{1,2}$ minimizes the Willmore energy for a Klein bottle immersed in $\mathbb{S}^3$ to any of several conjectural characterizations of the Lawson M\"obius band.