{
  "abstract": "Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature $K\\in (-1,0]$, and let $c\\in \\cT_S$ be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in $\\HH^3$ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if $h^*$ is a smooth metric on $S$ with curvature $K^*\\in (-\\infty, 0)$ and if $c\\in \\cT_S$, there exists a unique equivariant immersion of $S$ into $\\HH^3$ with third fundamental form $h^*$ and such that the pull-back of the conformal class at infinity by the Gauss map is $c$. Equivalently, given $h$ and $c\\in \\cT_S$, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given $h^*$ and $c$, there exists a unique pair $(E,u^*)$, where $E$ is a hyperbolic end with conformal structure at infinity $c$, and $u^*:S\\to E$ is an embedding inducing the third fundamental form $h^*$. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32300",
  "author_total": 1,
  "authors": [
    {
      "name": "Jean-Marc Schlenker"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9a75fyrgee5nx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32300",
  "fulltext_url": null,
  "id": "9a75fyrgee5nx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32300",
  "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"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214758818",
  "pdf_url": "https://arxiv.org/pdf/2609.32300",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9a75fyrgee5nx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32300",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32300"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Equivariant isometric immersions of surfaces in hyperbolic space",
  "title_text": "Equivariant isometric immersions of surfaces in hyperbolic space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9a75fyrgee5nx",
  "venue": "arXiv",
  "versions": []
}