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-1 for the energy-critical defocusing initial-value problem (i partial derivative(t) + Delta(g))u = u vertical bar u vertical bar(4), u(0) = phi, on hyperbolic space H-3.