Regular Lagrangians in Lefschetz fibrations
Abstract
We characterize regularity of Lagrangian submanifolds in Weinstein Lefschetz fibrations, establishing a conjecture of Giroux and Pardon. Our main result is the Weinstein analogue of a closed symplectic Lefschetz pencil result of Auroux, Mu\~noz, and Presas. As an application, given a Legendrian link in tight S3 and an exact filling which is part of an arboreal skeleton for the 4-ball, we build a Lefschetz fibration such that the image of the filling and all of its mutations are arcs in the base.
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