Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians

@misc{breen2026nearly,
  title = {{Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians}},
  author = {Joseph Breen and Agniva Roy and Luya Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08440},
  eprint = {2610.08440},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08440},
}

TY  - UNPB
TI  - Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians
AU  - Joseph Breen
AU  - Agniva Roy
AU  - Luya Wang
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08440
UR  - https://arxiv.org/abs/2610.08440
AB  - We study Stein structures on nearly Lefschetz fibrations, with applications to symplectic and Lagrangian submanifolds in Weinstein domains. We prove that, up to Weinstein deformation equivalence, all Lagrangian disks with Legendrian boundary in $4$-dimensional Weinstein domains are regular in the sense of Eliashberg-Ganatra-Lazarev. This settles a Weinstein analogue of the nearby Lagrangian conjecture for two-dimensional Lagrangian disks, and resolves part of a Lagrangian analogue of the Slice-Ribbon conjecture. Along the way, we show that positive allowable nearly Lefschetz fibrations are supported by canonical Stein structures and prove a corresponding quasiflexibility result in the planar case. This yields a generalization of work of Boileau-Orevkov which may be of independent interest. Finally, we give an explicit algorithm producing a Weinstein Kirby diagram from the nearly Lefschetz fibration structure of a multisection complement.
ER  -