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Rigidity of free boundary minimal annuli with elliptic-cylinder boundary image

Show that if a free boundary minimal surface N′ in the Euclidean 3-ball is embedded by first Steklov eigenfunctions and its boundary equals the intersection of the unit sphere with an elliptic cylinder, then N′ must be the critical catenoid.

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Background

In their Steklov limit analysis, the authors encounter a potential degeneration to a free boundary minimal surface whose boundary lies on an elliptic cylinder on the sphere.

Identifying the critical catenoid as the unique possibility would strengthen their gap arguments and simplify existence proofs.

References

We conjecture that the only such surface N′ is the critical catenoid; indeed, such a surface must have two boundary components, but it is not clear a priori whether the genus can be positive.

— 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 1.5 Extremal metrics on basic reflection surfaces and applications