On families of Lagrangian submanifolds

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 omega(t) are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Phi(t) of diffeomorphisms of X such that omega(t) = Phi(t)* (omega(o)). If L subset of X is a Lagrangian submanifold for (X, omega(0)), L-t = Phi(t)(-1)(L) is thus a Lagrangian submanifold for (X, omega(t)). Here we show that if we simply assume that L is compact and omega(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.