{
  "abstract": "We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8π$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,α}$ for every $α<\\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00501",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohameden Ahmedou"
    },
    {
      "name": "Ruben Jakob"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ck9aezvr5wakq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00501",
  "fulltext_url": null,
  "id": "ck9aezvr5wakq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00501",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7206214700",
  "pdf_url": "https://arxiv.org/pdf/2609.00501",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ck9aezvr5wakq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00501",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00501"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow",
  "title_text": "Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/ck9aezvr5wakq",
  "venue": "arXiv",
  "versions": []
}