{
  "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 $b \\in \\{1, 2, 3, 4, 5, 8\\}$. 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 $4π$, 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 $b \\geq 2$ 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 $b \\to \\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25787",
  "author_total": 2,
  "authors": [
    {
      "name": "Haoyu Chen"
    },
    {
      "name": "Braxton Osting"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9drvqz7yemq3c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25787",
  "fulltext_url": null,
  "id": "9drvqz7yemq3c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25787",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223060939",
  "pdf_url": "https://arxiv.org/pdf/2610.25787",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9drvqz7yemq3c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25787",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues",
  "title_text": "Computation of Free Boundary Minimal Tori via Extremal Steklov Eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/9drvqz7yemq3c",
  "venue": "arXiv",
  "versions": []
}