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
for
the energy-critical defocusing initial-value problem
on hyperbolic space
.
Keywords
global well-posedness, energy-critical defocusing NLS,
nonlinear Schrödinger equation, induction on energy