{
  "abstract": "We prove that, in every closed Riemannian three-manifold $(M^3,\\bar g)$, there exists a constant $C=C(M,\\bar g)>0$ such that every closed smoothly embedded minimal surface $Σ\\subset M$ satisfies $b_1(Σ;\\mathbb{Z}_2)\\le C\\bigl(\\operatorname{Ind}(Σ)+\\operatorname{Area}(Σ)\\bigr)$, establishing Song's additive genus-index-area conjecture. For closed connected orientable two-sided minimal immersions, we prove the universal inequality $γ(Σ)\\le8\\operatorname{Ind}(Σ)$ in positive Ricci curvature and the estimate $γ(Σ)\\le C(M,\\bar g)\\operatorname{Ind}(Σ)$ in positive scalar curvature. In the latter case, we also bound both area and total curvature by $C(M,\\bar g)(1+\\operatorname{Ind}(Σ))$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32001",
  "author_total": 1,
  "authors": [
    {
      "name": "Riccardo Caniato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6txpsg90jy4ks.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32001",
  "fulltext_url": null,
  "id": "6txpsg90jy4ks",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32001",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214729874",
  "pdf_url": "https://arxiv.org/pdf/2609.32001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6txpsg90jy4ks.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2609.32001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds",
  "title_text": "Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6txpsg90jy4ks",
  "venue": "arXiv",
  "versions": []
}