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Codimension-two realization for nonorientable surfaces via eigenvalue optimization

Show that every closed nonorientable surface admits a minimal embedding in S^4 as an S^2-doubling of area < 8π, and every compact nonorientable surface with boundary admits a free boundary minimal embedding in B^4 as a B^2-doubling of area < 2π.

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Background

The authors’ methods apply most strongly in the orientable, codimension-one setting; extending to nonorientable cases suggests moving to codimension two.

This would parallel their orientable realization and further broaden eigenvalue-based constructions.

References

Every closed nonorientable surface admits a minimal embedding in S4 as an S2-doubling, with area <8π. Likewise, every compact nonorientable surface with boundary admits a free boundary minimal embedding in \mathbb{B}4 as a B2-doubling, with area <2π.

— Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization (2402.13121 - Karpukhin et al., 20 Feb 2024) in Section 1.6 Equivariant optimization beyond orientable basic reflection surfaces