Least-area higher-genus non-orientable surfaces

Prove that, for every even integer \(m\geq3\), the least-area embedded non-orientable minimal surface of genus \(m\) bounded by a great circle is the surface \(\tilde{\xi}_{1,m-1}\).

Background

The paper constructs many embedded non-orientable minimal surfaces with boundary a great circle but establishes the least-area and index characterization only for Möbius bands. This question proposes the corresponding explicit minimizer for higher even non-orientable genera.

References

For $m$ an even integer at least $3$, show that the least area non-orientable embedded minimal surface of genus $m$ bounded by $C$ is the surface $\tilde{\xi}_{1,m-1}$ (constructed in ).

— On the Willmore energy of Möbius bands  (2609.26745 - Bernstein et al., 22 Sep 2026) in Question 8.5, Section 8 (Problems)

Do there exist non-orientable minimal equatorial fillings in higher dimensional spheres aside from the (singular) iterated suspensions of the ones known in \mathbb{S}3$? Are there any smoothly embedded such fillings?

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Questions