On families of Lagrangian submanifolds

Abstract

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ t of diffeomorphisms of X such that ω t =Φ t *(ω0). If L⊂X is a Lagrangian submanifold for (X,ω0), L t =Φ t -1(L) is thus a Lagrangian submanifold for (X,ω t ). Here we show that if we simply assume that L is compact and ω t | L is exact for every t, a family L t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure.