Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues
Abstract
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 ∈ {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 4π, 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 ≥ 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 4π as b → ∞.
Related
- Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
- Sharp upper bounds for Steklov eigenvalues on warped products
- Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
- Isodiametric-type upper bounds for Steklov eigenvalues
BibTeX
@misc{chen2026computation,
title = {{Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues}},
author = {Haoyu Chen and Braxton Osting},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25787},
eprint = {2610.25787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25787},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues
AU - Haoyu Chen
AU - Braxton Osting
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25787
UR - https://arxiv.org/abs/2610.25787
AB - 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 ∈ {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 4π, 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 ≥ 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 4π as b → ∞.
ER -