{
  "abstract": "By the paper \\cite{Pet2009upper}, \\cite{LiNan2026}, we know that there exists $C(n)>0$, for any complete compact or non-compact Riemannian n manifold $M$ with non-negative sectional curvature, without boundary and any $R>0$, \\begin{equation} R^{2-n}\\int_{B(p,R)} Scal \\le C(n). \\end{equation} Based on this result, we will prove that (1) There exists constant $C(n)$. If $M$ is a complete, n dimensional, non-compact Riemannian manifold with non-negative sectional curvature, then for any $p\\in M$, $R>0$, \\begin{equation} R^{2-n}\\int_{B(p,R)} Scal \\ dvol \\le C(n)(1-v(M)), \\end{equation} where $Scal$ is the scalar curvature and $v(M)=\\lim\\limits_{R\\to \\infty} \\frac{volB(p,R)}{volB(0,R)}$ is the asymptotic volume ratio. (2) There exists constant $C(n)$. If $M$ is a complete, n dimensional Riemannian manifold with sectional curvature $\\ge 1$, then \\begin{equation} \\int_M (Scal-n(n-1)) \\ dvol \\le C(n)(1-\\frac{vol(M)}{vol(S^n(1))}). \\end{equation}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29982",
  "author_total": 2,
  "authors": [
    {
      "name": "Yin Jiang"
    },
    {
      "name": "Nan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2cxd6xmjj7s9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29982",
  "fulltext_url": null,
  "id": "f2cxd6xmjj7s9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29982",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214403088",
  "pdf_url": "https://arxiv.org/pdf/2609.29982",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2cxd6xmjj7s9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29982",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29982"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below",
  "title_text": "Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/f2cxd6xmjj7s9",
  "venue": "arXiv",
  "versions": []
}