On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds

Using suitable modified energies, we study higher-order Sobolev norms' growth in time for the nonlinear Schrödinger equation (NLS) on a generic 2- or 3-dimensional compact manifold. In two dimensions, we extend earlier results that dealt only with cubic nonlinearities, and get polynomial-in-time bounds for any higher-order nonlinearities. In three dimensions, we prove that solutions to the cubic NLS grow at most exponentially, while for the subcubic NLS we get polynomial bounds on the growth of the H2 norm.