Index lower bound for minimal surfaces in S^3

Prove that any embedded closed minimal surface M in S^3 of genus γ satisfies ind(M) ≥ 2γ + 3, with equality for the Lawson surface ξ_{γ,1}.

Background

Kapouleas–Wiygul computed the index of the Lawson ξ_{γ,1} surfaces to be 2γ+3; the conjecture asserts these have minimal possible index in their genus class.

Establishing this would sharply characterize spectral instability across the moduli of minimal surfaces in S3.

References

If M is a closed minimal surface in S3 of genus γ, then \begin{align*} \mathrm{ind}(M) \geq \mathrm{ind}(\xi_{\gamma, 1}) = 2\gamma+3. \end{align*}

— 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