%0 Journal Article
%T Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr£żdinger Equations
%A N. Burq
%A P. Gerard
%A N. Tzvetkov
%J Mathematics
%D 2004
%I arXiv
%X We study nonlinear Schr\"odinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space $\R^3$ and the flat torus $\T^3$ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.
%U http://arxiv.org/abs/math/0409015v2