{
  "abstract": "Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\\geq 1$. For each prescribed volume fraction $s\\in(0,1)\\setminus\\{\\frac12\\}$, we prove that the isoperimetric region of volume $s \\operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two minimizers, a region $E$ and its complement $E^c$. As consequences, for every $m>0$, uniqueness holds for a generic set of metrics $g$ satisfying $m<\\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0<m<\\operatorname{Vol}_g(M)$. The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20790",
  "author_total": 1,
  "authors": [
    {
      "name": "Gongping Niu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jm6wh23nyr58.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20790",
  "fulltext_url": null,
  "id": "7jm6wh23nyr58",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20790",
  "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": "W7213612841",
  "pdf_url": "https://arxiv.org/pdf/2609.20790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jm6wh23nyr58.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension",
  "title_text": "Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/7jm6wh23nyr58",
  "venue": "arXiv",
  "versions": []
}