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PreprintCC BY 4.0arXiv

Nearby Lagrangians in the cotangent bundle of the Klein bottle

  • Yuan Gao
  • Weiwei Wu
  • Tianyu Yuan
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Abstract

We prove the nearby Lagrangian conjecture for the Klein bottle: every closed connected embedded exact Lagrangian submanifold of its cotangent bundle is Hamiltonian isotopic to the zero section by a compactly supported isotopy. The proof passes to the torus double cover and builds on the theorem of Dimitroglou Rizell, Goodman, and Ivrii for the two-dimensional torus. The main step is to construct an isotopy compatible with the covering involution. We achieve this using holomorphic sphere rulings in S2× S2 adapted to the zero section, allowing the construction to descend to the Klein bottle.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026nearby,
  title = {{Nearby Lagrangians in the cotangent bundle of the Klein bottle}},
  author = {Yuan Gao and Weiwei Wu and Tianyu Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01506},
  eprint = {2611.01506},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01506},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nearby Lagrangians in the cotangent bundle of the Klein bottle
AU  - Yuan Gao
AU  - Weiwei Wu
AU  - Tianyu Yuan
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01506
UR  - https://arxiv.org/abs/2611.01506
AB  - We prove the nearby Lagrangian conjecture for the Klein bottle: every closed connected embedded exact Lagrangian submanifold of its cotangent bundle is Hamiltonian isotopic to the zero section by a compactly supported isotopy. The proof passes to the torus double cover and builds on the theorem of Dimitroglou Rizell, Goodman, and Ivrii for the two-dimensional torus. The main step is to construct an isotopy compatible with the covering involution. We achieve this using holomorphic sphere rulings in S^2× S^2 adapted to the zero section, allowing the construction to descend to the Klein bottle.
ER  -