Weinstein analogue of the nearby Lagrangian conjecture for two-dimensional disks
Let be a -dimensional Weinstein domain, and let be a properly embedded Lagrangian disk with Legendrian boundary . Is regular in the sense of Eliashberg--Ganatra--Lazarev, possibly after Weinstein deformation of the ambient domain?
References
Primary source
Additional references
- Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians — arXiv — Joseph Breen, Agniva Roy, Luya Wang
Progress summary
A new preprint claims to settle the two-dimensional disk case in a four-dimensional setting, while broader versions of the conjecture remain open.
The problem asks whether the relevant two-dimensional Lagrangian disks in a four-dimensional Weinstein setting are regular. The newly reported work addresses this specific disk case rather than the broader nearby-Lagrangian conjecture.
October 2026 claimed disk-case result
Joseph Breen, Agniva Roy, and Luya Wang claim to prove regularity for these disks. Their preprint also develops nearly Lefschetz-fibration and Stein-structure methods and resolves part of a Lagrangian Slice–Ribbon analogue, so it constitutes a claimed resolution of the tracked disk problem.
Current status (as of October 2026): The specific two-dimensional disk statement is claimed proved by Breen, Roy, and Wang, but the claim remains unverified; broader nearby-Lagrangian conjectures remain open.
Sources
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