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

Abstract

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 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

$$\left(i{\partial }_t+{\Delta }_g\right)u=u|u{|}^4,u\left(0\right)=\varphi ,$$

on hyperbolic space ${ℍ}^3$.