{
  "abstract": "We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz--Minkowski three-space. On the hyperbolic plane, solutions of $Δu-2u=0$ give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein--Gordon equation $\\Box u+2u=0$, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13992",
  "author_total": 2,
  "authors": [
    {
      "name": "Márcio Batista"
    },
    {
      "name": "Iury Domingos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kjn30zccw5sh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13992",
  "fulltext_url": null,
  "id": "4kjn30zccw5sh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13992",
  "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": "W7213312314",
  "pdf_url": "https://arxiv.org/pdf/2609.13992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kjn30zccw5sh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Overdetermined equations and support functions on the hyperbolic and de Sitter planes",
  "title_text": "Overdetermined equations and support functions on the hyperbolic and de Sitter planes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4kjn30zccw5sh",
  "venue": "arXiv",
  "versions": []
}