  In this paper we present a method to study global regularity properties of
solutions of large-data critical Schrodinger equations on certain noncompact
Riemannian manifolds. We rely on concentration compactness arguments and a
global Morawetz inequality adapted to the geometry of the manifold (in other
words we adapt the method of Kenig-Merle to the variable coefficient case), and
a good understanding of the corresponding Euclidean problem (in our case the
main theorem of Colliander-Keel-Staffilani-Takaoka-Tao).
  As an application we prove global well-posedness and scattering in $H^1$ for
the energy-critical defocusing initial-value problem
(i\partial_t+\Delta_\g)u=u|u|^{4} on the hyperbolic space $H^3$.
