Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M .We prove global existence of strong H 1 solutions on M = S 3 and M = S 2 × S 1 as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre, Velo and Bourgain who treated the cases of the Euclidean space R 3 and the torus T 3 = R 3 /Z 3 respectively.The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.