Legendrian realization in convex Lefschetz fibrations and convex stabilizations

We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on partial derivative W, can be assumed to be Legendrian in partial derivative 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. 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 partial derivative W and partial derivative W' are contactomorphic