On the global well-posedness of energy-critical Schrödinger equations in curved spaces

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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 H¹ for the energy-critical defocusing initial-value problem (i∂ₜ+Δ)u=u|u|⁴ on the hyperbolic space H³.