Monotonicity of the maximized first Laplace eigenvalue across genus

Prove the strict inequality Λ1(M_{γ+1}) > Λ1(M_γ) for closed orientable surfaces, where Λ1(M) denotes the supremum of the area-normalized first Laplace eigenvalue over all Riemannian metrics on M.

Background

Petrides’ program reduces global existence of Λ1-maximizing metrics to showing strict growth of the conformal supremum with genus.

While non-strict monotonicity is known, attempts to prove strictness have gaps, and the inequality remains unsettled in general.

References

...to our knowledge the strict inequality g.mono remains open in general.

g.mono:

Λ1(Mγ+1)>Λ1(Mγ).\Lambda_1(M_{\gamma+1})>\Lambda_1(M_{\gamma}).

— 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 Existence theory for extremal metrics