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Lawson surfaces minimize area among embedded minimal surfaces of fixed genus

Prove that, for each genus γ, the minimum area A_γ among all embedded closed minimal surfaces in S^3 of genus γ is uniquely attained by the Lawson surface ξ_{γ,1}.

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Background

The conjecture is a special case of a broader Willmore-type minimization picture and has long guided intuition about the geometry of minimal surfaces in S3.

The authors’ constructions yield many near-minimizers but do not resolve the global minimization.

References

The minimum area A_γ among all minimal surfaces in S3 of genus γ is uniquely attained by the Lawson surface ξ_{γ, 1}.

— 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 Discussion and open questions