{
  "abstract": "We study the moduli space of complete maximal surfaces in pseudo-hyperbolic space $\\mathbf{H}^{2,2}$ with polygonal asymptotic boundary a lightlike $(n+4)$-gon in the Einstein universe $\\mathbf{Ein}^{1,2}$, up to the action of $\\mathsf{SO}_0(2,3)$. Importantly, this space $M_{n+4}(\\mathbf{H}^{2,2})$ is disconnected for $n\\geq 2$. We introduce a \\emph{turning invariant} $Θ$ that classifies the components in terms of the geometry of the boundary polygons in the Einstein universe $\\mathbf{Ein}^{1,2}$, which is related to \\emph{spherical polygons} in $\\mathbb{S}^2$. We provide a geometric parametrization of the moduli space $M_{n+4}(\\mathbf{H}^{2,2})$ via holomorphic data, namely a pair of polynomials $α$, $β$. These polynomials appear in relation to certain components of the complexified second fundamental form of the corresponding surface. We then relate the holomorphic data $α,β$ with the turning invariant $Θ$. This leads to characterizations of the \\emph{Hitchin component} of polygonal maximal surfaces studied by Tamburelli-Wolf.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22329",
  "author_total": 2,
  "authors": [
    {
      "name": "Parker Evans"
    },
    {
      "name": "Alex Moriani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/352pc56t6cxw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22329",
  "fulltext_url": null,
  "id": "352pc56t6cxw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22329",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222648256",
  "pdf_url": "https://arxiv.org/pdf/2610.22329",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/352pc56t6cxw7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22329",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22329"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons",
  "title_text": "Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/352pc56t6cxw7",
  "venue": "arXiv",
  "versions": []
}