{
  "abstract": "Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \\[ \\lim_{r\\to\\infty}\\frac1r\\int_{B_p(r)}\\operatorname{Scal}\\,d V =8π\\bigl(χ(M)-V_M\\bigr)\\le8π(1-V_M), \\] where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $χ(M)\\in\\{0,1\\}$. Thus positive asymptotic volume ratio gives the value $8π(1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8πχ(N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26717",
  "author_total": 1,
  "authors": [
    {
      "name": "Qixuan Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9nmbxhrxpdpbn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26717",
  "fulltext_url": null,
  "id": "9nmbxhrxpdpbn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26717",
  "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": "W7214077416",
  "pdf_url": "https://arxiv.org/pdf/2609.26717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9nmbxhrxpdpbn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.26717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Scalar curvature growth on nonnegatively curved three-manifolds",
  "title_text": "Scalar curvature growth on nonnegatively curved three-manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/9nmbxhrxpdpbn",
  "venue": "arXiv",
  "versions": []
}