{
  "abstract": "For every pair of integers $3\\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $σ_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\\langle X,ν\\rangle=-σ_k$, where $X$ is the position vector and $ν$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01346",
  "author_total": 3,
  "authors": [
    {
      "name": "Haoxuan Cheng"
    },
    {
      "name": "Junqi Lai"
    },
    {
      "name": "Guoxin Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3srg2151mzrzt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01346",
  "fulltext_url": null,
  "id": "3srg2151mzrzt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01346",
  "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": "W7206183465",
  "pdf_url": "https://arxiv.org/pdf/2609.01346",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3srg2151mzrzt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01346",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01346"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Singular Rotational Self-Similar Tori for Odd $σ_k$-Curvature Flows",
  "title_text": "Singular Rotational Self-Similar Tori for Odd σ_k-Curvature Flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/3srg2151mzrzt",
  "venue": "arXiv",
  "versions": []
}