{
  "abstract": "We show that among Möbius bands in $\\mathbb{S}^3$ bounded by a great circle, the minimal Willmore energy is realized by an embedded minimal Möbius band with Morse index two. To prove this, we introduce a $2$-parameter ``canonical family\" associated to any non-orientable surface in $\\mathbb{S}^3$ with boundary a great circle and apply a min-max argument. The canonical family detects the Euler number of the surface and is inspired by the $5$-parameter family discovered by F.C. Marques and A. Neves detecting the genus of an orientable surface in $\\mathbb{S}^3$. For $\\mathbb{Z}_2$-invariant Klein bottles, this reduces R. Kusner's 1989 conjecture that $τ_{1,2}$ minimizes the Willmore energy for a Klein bottle immersed in $\\mathbb{S}^3$ to any of several conjectural characterizations of the Lawson Möbius band.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26745",
  "author_total": 2,
  "authors": [
    {
      "name": "Jacob Bernstein"
    },
    {
      "name": "Daniel Ketover"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2kp8ajhp2049c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26745",
  "fulltext_url": null,
  "id": "2kp8ajhp2049c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26745",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214114382",
  "pdf_url": "https://arxiv.org/pdf/2609.26745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2kp8ajhp2049c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Willmore energy of Möbius bands",
  "title_text": "On the Willmore energy of Möbius bands",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2kp8ajhp2049c",
  "venue": "arXiv",
  "versions": []
}