NONLINEAR SCHRODINGER EQUATION ON FOUR-DIMENSIONAL COMPACT MANIFOLDS

We prove two new results about the Cauchy problem in the energy space for nonlinear Schrodinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy data in the Hamiltonian case. Both results are based on new multilinear Strichartz-type estimates for the Schrodinger group.