{
  "abstract": "We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension $n\\ge3$, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On $\\mathbb T^3$, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on $\\mathbb T^3$ admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18428",
  "author_total": 1,
  "authors": [
    {
      "name": "Hoan Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fh3522t2ygtc0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18428",
  "fulltext_url": null,
  "id": "fh3522t2ygtc0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18428",
  "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"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7203894938",
  "pdf_url": "https://arxiv.org/pdf/2608.18428",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fh3522t2ygtc0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.18428",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18428"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Minimal foliations, codimension-one stable norms, and a question of Bangert",
  "title_text": "Minimal foliations, codimension-one stable norms, and a question of Bangert",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fh3522t2ygtc0",
  "venue": "arXiv",
  "versions": []
}