{
  "abstract": "We study the geometry and topology of transversely vanishing eigenforms of the curl operator on $3$-manifolds, placing them in the broader framework of near-contact forms. We prove a Gray-type stability theorem for the kernels of near-contact forms, which we call near-contact structures, and derive a local neighborhood theorem around their singularities. As applications, we obtain a stability result for the contact topology of curl eigenforms, a classification of near-contact structures in terms of formal homotopical data, and a proof of the existence of contact structures on closed orientable 3-manifolds based on the spectral theory of the curl operator. We also study the symplectization of curl eigenforms as a possible bridge to symplectic dynamics. Finally, we characterize those transversely vanishing $1$-forms which are curl eigenforms for some smooth Riemannian metric. We deduce a generalization of the classical correspondence, due to Etnyre and Ghrist, between Reeb-like fields of contact forms and nowhere-vanishing rotational Beltrami fields to the setting of transversely vanishing vector fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36212",
  "author_total": 2,
  "authors": [
    {
      "name": "Robert Cardona"
    },
    {
      "name": "Willi Kepplinger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6hn7pdwj5kr8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36212",
  "fulltext_url": null,
  "id": "6hn7pdwj5kr8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36212",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7224116616",
  "pdf_url": "https://arxiv.org/pdf/2610.36212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6hn7pdwj5kr8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Contact topology and intrinsic characterization of transversely vanishing curl eigenforms",
  "title_text": "Contact topology and intrinsic characterization of transversely vanishing curl eigenforms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6hn7pdwj5kr8f",
  "venue": "arXiv",
  "versions": []
}