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 H1 for the energy-critical defocusing initial-value problem on hyperbolic space ℍ3.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Applied Mathematics
- Energy-critical defocusing nls
- Global well-posedness
- Induction on energy
- Nonlinear schrödinger equation