Willmore energy and minimal fillings

Abstract

For each $n\ge 4$, we show that the infimal Willmore energy of a non-orientable surface of odd genus immersed in ${\mathbb{S}}^n$ is attained by an index $(n+1)$ minimal surface in this class. For $n=4$, we obtain in addition the analogous conclusion for surfaces with non-zero Euler number. The proof uses the fact that such surfaces represent a non-trivial cobordism class together with an observation of M. Gromov relating the conformal area in ${\mathbb{S}}^n$ to minimal fillings in the hyperbolic space ${\mathbb{H}}^{n+1}$.