A multidimensional Birkhoff Theorem for time-dependent Tonelli Hamiltonians

Overview & Synopsis

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