Engouffrement symplectique et intersections lagrangiennes

Let }L _t{,t ∈ 0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension ≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L _t{ is conjugate to a path of exact submanifolds inT ^*L_o. This impliesL _t must intersectL _o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property of symplectic structures which is stated in the first part of this work.