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

Abstract

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 and Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (a theorem of Colliander, Keel, Staffilani, Takaoka and Tao). As an application we prove global well-posedness and scattering in H[superscript 1] for the energy-critical defocusing initial-value problem (i∂t + Δ[subscript g])u = u|u|[superscript 4], u(0) = ϕ, on hyperbolic space ℍ[superscript 3].