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

We study nonlinear Schrodinger 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 x S' 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. (c) 2005 Elsevier SAS.