{
  "abstract": "We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\\operatorname{Ric}_g\\ge (n-1)g$ and the scalar curvature $R_g\\ge n(n-1)(1+\\varepsilon)$, then its volume satisfies $$\\lvert N\\rvert_g \\le \\frac{1}{\\sqrt{1+n\\varepsilon}}\\lvert\\mathbb S^n\\rvert. $$ In fact, assuming only $\\mathrm{Ric}_g\\ge(n-1)g$, we can prove that $$\\frac{|N|_g}{\\left|\\mathbb{S}^n\\right|} \\le \\frac{1}{|N|_g} \\int_N\\left(\\frac{R_g}{n-1}-(n-1)\\right)^{-\\frac{1}{2}} d \\mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \\cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19196",
  "author_total": 1,
  "authors": [
    {
      "name": "Kwok-Kun Kwong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27zw896bn2510.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19196",
  "fulltext_url": null,
  "id": "27zw896bn2510",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19196",
  "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.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7204216770",
  "pdf_url": "https://arxiv.org/pdf/2608.19196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27zw896bn2510.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.19196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An improved volume bound under Ricci and scalar curvature lower bounds",
  "title_text": "An improved volume bound under Ricci and scalar curvature lower bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/27zw896bn2510",
  "venue": "arXiv",
  "versions": []
}