Willmore Energy Estimates for Klein Bottles

@misc{ni2026willmore,
  title = {{Willmore Energy Estimates for Klein Bottles}},
  author = {Ruijie Ni and Peng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36519},
  eprint = {2609.36519},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36519},
}

TY  - UNPB
TI  - Willmore Energy Estimates for Klein Bottles
AU  - Ruijie Ni
AU  - Peng Wang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36519
UR  - https://arxiv.org/abs/2609.36519
AB  - In this note, we provide an estimate for the Willmore energy of Klein bottles in ℝ^N, building on the ideas of Li-Yau and Montiel-Ros for $2$-tori. In particular, we prove that the Willmore energy 𝒲(ϕ)> 6π for any conformal branched immersion ϕ:𝕂^2_b→ ℝ^N, where 𝕂^2_b is a flat Klein bottle with 0.350≲ b ≲ 0.755. This confirms partially a conjecture of Kusner. Moreover, for a flat immersion ψ:𝕂^2_b→ ℝ^N, we derive a lower bound of 𝒲(ψ), which confirms a conjecture of Hirsch-Mäder-Baumdicker for flat Klein bottles in ℝ^N. We also construct a smooth family of flat Klein bottles in ℝ^5, with Willmore energy between 6.912π and 8π.
ER  -