A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians

Let M be a closed and connected manifold, H : T*M x R/Z -> R a Tonelli 1-periodic Hamiltonian and L subset of T * M a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if L is invariant by the time- one map of H, then L is a graph over M. An interesting consequence in the autonomous case is that in this case, L is invariant by all the time t maps of the Hamiltonian flow of H.