{
  "abstract": "Let $X$ be a genus $g\\geq 0$ compact Riemann surface that admits an antiholomorphic involution $ι$ with non empty fixed-point set, and let $D = \\{p_1,\\dots,p_k\\}\\subset \\text{Fix}(ι)$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from $\\mathbb{C}$ to the interior of an ideal polygon in $\\mathbb{H}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map \\[ h:X\\backslash D \\to \\mathbb{H}^2 \\] whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of $h$ tends to an ideal polygon near each end. The resulting $h$ covers harmonic maps obtained by taking the $\\mathbb{H}^2$ factors of horizontal catenoids in $\\mathbb{H}^2 \\times \\mathbb{R}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18297",
  "author_total": 3,
  "authors": [
    {
      "name": "Qiongling Li"
    },
    {
      "name": "Jinsong Liu"
    },
    {
      "name": "Yihui Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hd8pjdbycaw5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18297",
  "fulltext_url": null,
  "id": "0hd8pjdbycaw5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18297",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213528933",
  "pdf_url": "https://arxiv.org/pdf/2609.18297",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hd8pjdbycaw5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.18297",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18297"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics",
  "title_text": "Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0hd8pjdbycaw5",
  "venue": "arXiv",
  "versions": []
}