Legendrian submanifolds from Bohr–Sommerfeld covers of monotone Lagrangian tori

By a result due to Ziltener, there exist no closed embedded Bohr–Sommerfeld Lagrangians inside $\mathbb{C}P^n$ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr–Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey–Lawson special Lagrangian cone), and compute the associated Chekanov–Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. Along the way, we also compute bilinearised Legendrian contact homology of a general Legendrian surface in the standard contact vector space when all Reeb chords are of positive degree, as well as the augmentation variety in the case of tori.

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Received 19 June 2019

Accepted 12 February 2021

Published 16 July 2024