{
  "abstract": "Let $\\{Σ_a\\}$, $a\\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\\subset\\mathbb{S}^3$, $R(a)>π/2$. We prove that $R$ is real-analytic, tends to $π/2$ at both ends, and therefore folds: it has an interior maximum $R_*>π/2$ and is not injective. Hence each $B(ρ)$ with $π/2<ρ<R_*$ contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of $R$ the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally invariant, reflection-even field not induced by any Killing field of $\\mathbb{S}^3$ preserving the ball; its nullity is at least three. These degenerate annuli form a nonempty discrete set; off it, the rotationally invariant even nullity vanishes. Those sitting at a maximizer of $R$, in the largest cap $B(R_*)$, are called degenerate annuli of maximal cap radius; each such annulus is also a strict local area maximizer in the family, since area and $R$ have the same critical points. Thus the hypothesis that all Jacobi fields of an embedded free boundary minimal annulus in a spherical cap are Killing-induced, used in the Naff--Zhu continuity approach to uniqueness, fails for some radius $R>π/2$. The exact identity $\\dim K_0^{ev}(Σ_a)=\\mathbf{1}_{\\{R'=0\\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\\partial_ηφ_a-\\operatorname{ct}_κ(r(a))φ_a=r'(a)A_a(η,η)$, a Robin defect identity, which requires of the ambient only that the barrier family be umbilic and which extends to capillary boundary conditions at constant contact angle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23534",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander Pigazzini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4gy63tmqnbtcx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23534",
  "fulltext_url": null,
  "id": "4gy63tmqnbtcx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23534",
  "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": "W7171521626",
  "pdf_url": "https://arxiv.org/pdf/2607.23534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4gy63tmqnbtcx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2607.23534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere",
  "title_text": "A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4gy63tmqnbtcx",
  "venue": "arXiv",
  "versions": []
}