{
  "abstract": "Gromov conjectured that for a compact Riemannian manifold $X$ with boundary, the total mean curvature $\\int_{\\partial X} H$ is bounded above by a constant depending only on the intrinsic geometry of $\\partial X$ and a lower bound on the scalar curvature of $X$. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions $n \\ge 3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01189",
  "author_total": 1,
  "authors": [
    {
      "name": "Christian Baer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7nw7ry4ggcyjn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01189",
  "fulltext_url": null,
  "id": "7nw7ry4ggcyjn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01189",
  "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": "W7206202728",
  "pdf_url": "https://arxiv.org/pdf/2609.01189",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7nw7ry4ggcyjn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01189",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01189"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture",
  "title_text": "The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/7nw7ry4ggcyjn",
  "venue": "arXiv",
  "versions": []
}