{
  "abstract": "For a smooth, connected, oriented surface $Σ$, topologically a sphere, immersed in the homogeneous space $E(κ,τ)$ with $τ\\ne0$, we show that a one-sided Cauchy--Riemann-type inequality on the original Abresch--Rosenberg differential $\\mathcal{Q}_{AR}$ -- much weaker than requiring $\\mathcal{Q}_{AR}$ to be holomorphic -- already forces the mean curvature $H$ to be constant. The proof is uniform across every $E(κ,τ)$, including the round sphere, and combines the Bers--Vekua similarity principle with the Poincaré-Hopf index formula for line fields; it is logically independent of the algebraic argument used, in the companion paper~\\cite{AlencarRosenberg2026}, to characterize CMC immersions by holomorphy of $\\mathcal{Q}_{AR}$ alone. We also show that the topological hypothesis is sharp: on every Berger sphere, including the round one, there exist compact non-CMC tori -- the preimages under the Hopf-type submersion $π:E(κ,τ)\\to M^2(κ)$ of simple closed curves in the base with nonconstant geodesic curvature -- satisfying the same inequality with a constant bound. As an elementary consequence of the classical Poincaré-Hopf theorem for line fields, we record that every closed surface on which $\\det S<0$ everywhere -- a condition satisfied, in particular, by every Hopf tube, where $\\det S\\equiv-τ^2$ -- is a torus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23227",
  "author_total": 2,
  "authors": [
    {
      "name": "Hilário Alencar"
    },
    {
      "name": "Harold Rosenberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42gz5a2nsv0ca.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23227",
  "fulltext_url": null,
  "id": "42gz5a2nsv0ca",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23227",
  "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": "W7214022981",
  "pdf_url": "https://arxiv.org/pdf/2609.23227",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42gz5a2nsv0ca.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23227",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23227"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spheres under a Cauchy-Riemann inequality in $E(κ,τ)$, and non-CMC Hopf tori",
  "title_text": "Spheres under a Cauchy-Riemann inequality in E(κ,τ), and non-CMC Hopf tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/42gz5a2nsv0ca",
  "venue": "arXiv",
  "versions": []
}