{
  "abstract": "We study the diffusion and reaction effects of Bartnik's quasi-spherical equation to deform boundary mean curvature in fill-in problems with scalar curvature bounded below. The diffusion effect yields an explicit $L^p$-to-$L^\\infty$ estimate for $\\partial_tu=u^2Δu$, thereby extending the known upper bound for the minimum boundary mean curvature of fill-ins to a quantitative upper bound for its harmonic mean. For spin fill-ins, this bound is explicit and involves only coarse intrinsic boundary data. By introducing an absorbing reaction term, we also construct a deformation that transforms any nonnegative initial mean curvature into a terminal mean curvature with a uniform positive lower bound. For Gromov's conjecture on total mean curvature, this reduces the $H\\geq 0$ case to Theorem A of Frenck, Hanke, and Hirsch \\cite{FHH}, which assumes $H\\geqκ>0$. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13999",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenlong Wang"
    },
    {
      "name": "Guodong Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8zkh72ynn2afy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13999",
  "fulltext_url": null,
  "id": "8zkh72ynn2afy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13999",
  "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": "W7213311339",
  "pdf_url": "https://arxiv.org/pdf/2609.13999",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8zkh72ynn2afy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13999",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13999"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Diffusion and reaction in the quasi-spherical equation: mean curvature deformations for fill-Ins",
  "title_text": "Diffusion and reaction in the quasi-spherical equation: mean curvature deformations for fill-Ins",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8zkh72ynn2afy",
  "venue": "arXiv",
  "versions": []
}