{
  "abstract": "In this note, we provide an estimate for the Willmore energy of Klein bottles in $\\mathbb R^N$, building on the ideas of Li-Yau and Montiel-Ros for $2$-tori. In particular, we prove that the Willmore energy $\\mathcal{W}(ϕ)> 6π$ for any conformal branched immersion $ϕ:\\mathbb{K}^2_b\\rightarrow \\mathbb R^N$, where $\\mathbb{K}^2_b$ is a flat Klein bottle with $0.350\\lesssim b \\lesssim 0.755$. This confirms partially a conjecture of Kusner. Moreover, for a flat immersion $ψ:\\mathbb{K}^2_b\\rightarrow \\mathbb R^N$, we derive a lower bound of $\\mathcal{W}(ψ)$, which confirms a conjecture of Hirsch-Mäder-Baumdicker for flat Klein bottles in $\\mathbb R^N$. We also construct a smooth family of flat Klein bottles in $\\mathbb R^5$, with Willmore energy between $6.912π$ and $8π$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36519",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruijie Ni"
    },
    {
      "name": "Peng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxwntx0bka6xq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36519",
  "fulltext_url": null,
  "id": "cxwntx0bka6xq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36519",
  "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": "W7214975726",
  "pdf_url": "https://arxiv.org/pdf/2609.36519",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxwntx0bka6xq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36519",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36519"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Willmore Energy Estimates for Klein Bottles",
  "title_text": "Willmore Energy Estimates for Klein Bottles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/cxwntx0bka6xq",
  "venue": "arXiv",
  "versions": []
}