Characterization and area monotonicity in the Z2-symmetric genus-γ family

Establish that (1) the Z2-symmetric embedded minimal surface M_{γ,0} is the unique minimal surface with its symmetry group and genus and is congruent to the Lawson surface ξ_{γ,1}; and (2) the areas satisfy area(M_{γ,a+1}) ≥ area(M_{γ,a}) for all integers a with 0 ≤ a ≤ ⌊γ/2⌋.

Background

The paper constructs large Z2-symmetric families of S2-doublings in S3 and suggests a hierarchy that could recover Lawson’s family at one endpoint.

Proving uniqueness and monotonicity would, in particular, imply Lawson area-minimization within these symmetry classes.

References

(1) M_{γ, 0} is the unique minimal surface with its symmetry group and genus, hence is congruent to ξ{γ,1}. (2) area(M{γ, a+1}) ≥ area(M_{γ, a}) for each a ∈ {0, …, ⌊ γ/2⌋}.

— Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization  (2402.13121 - Karpukhin et al., 2024) in Section 1.6 Discussion and open questions

The same conclusion is expected under the subgroup $G$ of order $8k$, which consists of those symmetries preserving the two sides of the surface, see Remark~4.6 and Lemma~3.11\,(ii). As long as this remains open, Theorem~\ref{thm:index-Lawson} covers a potentially larger class of surfaces.

— Index and nullity of minimal surfaces via representation theory  (2610.03023 - Karpukhin et al., 2 Oct 2026) in Introduction, Remark following Theorem \ref{thm:index-Lawson}