{
  "abstract": "For immersed surfaces in the homogeneous spaces $E(κ,τ)$ with $τ\\ne0$, we prove that the original Abresch--Rosenberg differential $[2(H+iτ)p-(κ-4τ^2)A^2]\\,dζ^2$ is holomorphic if and only if the mean curvature $H$ is constant; here $p\\,dζ^2$ is the Hopf differential and $A=\\langleξ,\\partial_ζ\\rangle$. The result is local: no completeness, compactness or analyticity is assumed, and there are no exceptional surfaces. Where $dH\\ne0$, holomorphy forces the shape operator into the normal form $\\bigl(\\begin{smallmatrix}0&τ\\\\τ&2H\\end{smallmatrix}\\bigr)$ in the frame $(\\nabla H/|\\nabla H|,J\\nabla H/|\\nabla H|)$. The round case $κ=4τ^2$ is settled by an elementary argument; for $κ\\ne4τ^2$ the structure equations become a rational system whose solutions are algebraic, and an embedding into a field of Puiseux series excludes every branch. In $M^2(κ)\\times\\mathbb{R}$, $κ\\ne0$, where the statement fails, we show that a non-CMC surface with holomorphic original differential has nowhere vanishing $dH$ and constant angle function, and belongs to an explicit family; in particular, closed surfaces with holomorphic original differential are CMC there. We also show that the Hopf-cylinder alternative in a theorem of Dorfmeister, Inoguchi and Kobayashi contains no non-CMC surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41197",
  "author_total": 2,
  "authors": [
    {
      "name": "Hilário Alencar"
    },
    {
      "name": "Harold Rosenberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4q0q1ggcbky3e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41197",
  "fulltext_url": null,
  "id": "4q0q1ggcbky3e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41197",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41197",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4q0q1ggcbky3e.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41197",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41197"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Only constant mean curvature surfaces in $E(κ,τ)$, $τ\\ne0$, have a holomorphic original Abresch-Rosenberg differential",
  "title_text": "Only constant mean curvature surfaces in E(κ,τ), τ≠0, have a holomorphic original Abresch-Rosenberg differential",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/4q0q1ggcbky3e",
  "venue": "arXiv",
  "versions": []
}