A multidimensional birkhoff theorem for time-dependent tonelli hamiltonians

Abstract

Let $M$ be a closed and connected manifold, $H:T^{\ast }M\times \mathbb{R}/\mathbb{Z}\to \mathbb{R}$ a Tonelli $1$-periodic Hamiltonian and $\mathcal{L}\subset T^{\ast }M$ 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 $H$.