Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues

Fraser and Schoen showed that the first Steklov eigenfunctions of a metric maximizing the length-normalized first eigenvalue of a surface with boundary define a free boundary minimal immersion into a Euclidean ball. We develop a method for this extremal problem on surfaces of genus one with $b$ boundary components and use it to compute free boundary minimal tori for $b\in \{1,2,3,4,5,8\}$. By uniformization, the problem is a nonsmooth maximization over the modulus of a flat torus, the centers and radii of $b$ removed disks, and a boundary weight. The eigenvalues are computed with doubly periodic layer potentials, all derivatives follow from a single formula for the underlying pencil of boundary integral operators, and the coalescence of eigenvalues at a maximizer is handled by a two-stage nonsmooth method. The Lagrange multiplier matrix that certifies first-order stationarity rescales the eigenfunctions into the free boundary minimal immersion. For $b=1$, the supremum is the Fraser–Schoen bound $4\pi$, attained by branched double covers of the disk; the embedded torus with connected boundary of Carlotto, Franz, and Schulz is a saddle point, which answers their question whether it maximizes the eigenvalue in the negative. For $b\ge 2$ the largest values correspond to free boundary minimal tori in $\mathbb{B}$^5 and $\mathbb{B}$^6; the surfaces of Folha, Pacard, and Zolotareva, two disks joined by a central neck and $b$ equatorial necks, are maximizers within their symmetry class but saddle points of the full problem, with values increasing toward $4\pi$ as $b\to \mathrm{\infty }$.