{
  "abstract": "We study the sharp homogeneous Dirichlet Gagliardo--Nirenberg quotient on a closed Riemannian manifold with holes of small capacity and connected complements. For each fixed exponent $2<q<2^*$, the nonnegative $L^2$-normalized extremal exists and is unique when the first Dirichlet eigenvalue $λ_D$ is sufficiently small. If the ambient volume is $V$ and $β=n(q-2)/2$, the sharp constant satisfies $C_D=V^{1-q/2}λ_D^{-β/2}(1+O(λ_D))$. For the normalized first eigenfunction $φ_D$, we obtain $\\|u_D-φ_D\\|_{H^1}=O(λ_D^{3/2})$ and $C_D/W_{M_D}(φ_D)=1+O(λ_D^2)$, together with a quantitative stability estimate in terms of the relative deficit. The constants depend only on the ambient manifold and the exponent. On general compact manifolds with Dirichlet boundary conditions, we prove compactness above the Euclidean sharp constant and describe concentration at equality. Positive scalar curvature at one interior point implies attainment. For the $L^2$-critical nonlinear Schrödinger energy, the mass threshold for negative energy tends to zero as the capacity of the holes tends to zero, whereas the threshold for unboundedness below is unchanged.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32173",
  "author_total": 2,
  "authors": [
    {
      "name": "Mayukh Mukherjee"
    },
    {
      "name": "Utsab Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8m7e81ktsxwbk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32173",
  "fulltext_url": null,
  "id": "8m7e81ktsxwbk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32173",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223690460",
  "pdf_url": "https://arxiv.org/pdf/2610.32173",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8m7e81ktsxwbk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32173",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32173"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Gagliardo--Nirenberg extremals on manifolds with small holes",
  "title_text": "Gagliardo--Nirenberg extremals on manifolds with small holes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/8m7e81ktsxwbk",
  "venue": "arXiv",
  "versions": []
}