#### Vol. 5, No. 4, 2012

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

### Alexandru D. Ionescu, Benoit Pausader and Gigliola Staffilani

Vol. 5 (2012), No. 4, 705–746
##### 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

 $\left(i{\partial }_{t}+{\Delta }_{g}\right)u=u|u{|}^{4},\phantom{\rule{1em}{0ex}}u\left(0\right)=\varphi ,$

on hyperbolic space ${ℍ}^{3}$.

##### Keywords
global well-posedness, energy-critical defocusing NLS, nonlinear Schrödinger equation, induction on energy
Primary: 35Q55