Legendrian Realization in Convex Lefschetz Fibrations and Convex Stabilizations

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n + 2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W , which we call convex open book, induced by a compact convex Lefschetz fibration on W carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on W . Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W , any simply connected embedded Lagrangian submanifold of a page in a convex open book on ∂W can be assumed to be Legendrian in ∂W with the induced contact structure. This can be thought as the extension of Giroux’s Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on W yields a compact convex Lefschetz fibration on a Liouville domain W ′ which is exact symplectomorphic to a positive expansion of W . In particular, with the induced structures ∂W and ∂W ′ are contactomorphic.