Area and Willmore minimization by the constructed surfaces

Establish that, for every odd integer m≥3, the surface \tilde{\xi}_{1,m} minimizes area and Willmore energy among non-orientable minimal surfaces with boundary a great circle and genus m+1.

Background

The surfaces \tilde{\xi}_{k,m} constructed in the paper have boundary a great circle, explicitly controlled genus, and symmetry. Their variational optimality is not proved; the authors propose an analogue of a long-standing conjecture concerning the orientable Lawson surfaces.

References

We have the following non-orientable analog to R. Kusner's long-standing conjecture for the Lawson surfaces \xi_{1,g} in \mathbb{S}3: For each odd m\in\mathbb{N} with m\geq 3, the surface \tilde{\xi}_{1,m} minimizes area and Willmore energy among non-orientable minimal surfaces with boundary a great circle and genus m+1.

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