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PreprintCC BY 4.0arXiv

On the Willmore energy of Möbius bands

  • Jacob Bernstein
  • Daniel Ketover
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Abstract

We show that among Möbius bands in 𝕊3 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 𝕊3 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 𝕊3. For ℤ2-invariant Klein bottles, this reduces R. Kusner's 1989 conjecture that τ1,2 minimizes the Willmore energy for a Klein bottle immersed in 𝕊3 to any of several conjectural characterizations of the Lawson Möbius band.

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Cite

BibTeXRIS
BibTeX
@misc{bernstein2026willmore,
  title = {{On the Willmore energy of Möbius bands}},
  author = {Jacob Bernstein and Daniel Ketover},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26745},
  eprint = {2609.26745},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26745},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Willmore energy of Möbius bands
AU  - Jacob Bernstein
AU  - Daniel Ketover
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26745
UR  - https://arxiv.org/abs/2609.26745
AB  - We show that among Möbius bands in 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 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 𝕊^3. For 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 𝕊^3 to any of several conjectural characterizations of the Lawson Möbius band.
ER  -