Global optimality of the computed higher-boundary configurations

Determine whether the numerically identified configurations with two through five boundary components attain the global extremal values \(\sigma^*(1,b)\) for the genus-one normalized first Steklov eigenvalue problem.

Background

The paper computes candidate maximizers of the normalized first Steklov eigenvalue for genus-one surfaces with several boundary components. For b=2,…,5b=2,\dots,5, the reported configurations have large values and are local maximizers in some cases, but the authors only establish that these values exceed lower bounds obtained from genus-zero problems and remain below the Fraser–Schoen upper bound. Consequently, the computations do not determine whether these candidates realize the global supremum σ∗(1,b)\sigma^*(1,b).

References

For b=2,\dots,5 they lie above the lower bounds \sigma(1,b)>\sigma^(0,b+1)=12.01, 13.67, 14.47, 15.43 that follow from the genus-zero values of \textcite[Table~2]{Kao_2020} and the gluing inequality of Theorem~1.3, and below the Fraser--Schoen bound 2\pi(b+1); whether they attain \sigma*(1,b) we cannot decide.

— Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues  (2610.25787 - Chen et al., 19 Oct 2026) in Section 4, subsection “Maximizers and saddle points”