{
  "abstract": "We study regular two-dimensional free boundary minimal $T$-surfaces in the unit ball $\\B^n$, with six faces, four triple junction arcs, and one tetrahedral vertex. A conformal minimal immersion of the standard flat $T$-cone, with the prescribed matching of junction traces, has the flat cone as its image. We also obtain rigidity criteria that do not require matching conformal parametrizations. In the disk-type class, two opposite planar faces suffice. More generally, it is enough that a natural junction invariant vanish on both junction edges of two opposite faces. For variations preserving the junction identifications and keeping the outer boundary on the sphere, we compute \\[ \\operatorname{ind}(TC)=n, \\qquad \\operatorname{nul}(TC)=3(n-2). \\] The normal kernel consists exactly of the normal components of ambient rotations, and we compute the associated coupled harmonic Steklov spectrum. Every regular free boundary minimal $T$-surface with six faces has index at least $n$, with equality if and only if it is the flat $T$-cone; the same characterization holds for ambient variations. Finally, the flat cone is locally rigid, modulo orthogonal transformations, among sufficiently small compatible facewise $C^2$ free boundary minimal deformations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.27776",
  "author_total": 1,
  "authors": [
    {
      "name": "Elham Matinpour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/esmevbvea1wdj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.27776",
  "fulltext_url": null,
  "id": "esmevbvea1wdj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.27776",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7162657923",
  "pdf_url": "https://arxiv.org/pdf/2605.27776",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/esmevbvea1wdj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2605.27776",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.27776"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rigidity and Morse index of free boundary minimal $T$-surfaces in the unit ball",
  "title_text": "Rigidity and Morse index of free boundary minimal T-surfaces in the unit ball",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/esmevbvea1wdj",
  "venue": "arXiv",
  "versions": []
}