Nonlinear Schr\"odinger equation on four-dimensional compact manifolds

We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere. The second one provides local wellposedness for quadratic nonlinearities in the case of zonal data on the sphere. Both results are based on new multilinear Strichartz-type estimates for the Schroedinger group.