On the Global Well-posedness of Energy-critical Schrödinger Equations in Curved Spaces

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrödinger 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 [40] to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-KeelStaffilani-Takaoka-Tao [21]). As an application we prove global well-posedness and scattering in H for the energycritical defocusing initial-value problem (i∂t +∆g)u = u|u| , u(0) = φ, on the hyperbolic space H.