Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow

In this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schr\"odinger equation with a convolution potential and for a nonlinear beam equation. We also prove $H^s$ stability of the ground state in NLS equation. The proof is based on a normal form procedure.