Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class

Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the cotangent space $T^{\ast }M$, strictly convex and superlinear in the momentum variables, we prove uniqueness of certain ergodic invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with rotation vector $\rho$. This result extends generically to the $C^0$-closure of KAM tori.