## Analysis & PDE

• Anal. PDE
• Volume 5, Number 4 (2012), 705-746.

### 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 $H1$ for the energy-critical defocusing initial-value problem

$( i ∂ t + Δ g ) u = u | u | 4 , u ( 0 ) = ϕ ,$

on hyperbolic space $ℍ3$.

#### Article information

Source
Anal. PDE, Volume 5, Number 4 (2012), 705-746.

Dates
Accepted: 1 April 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731243

Digital Object Identifier
doi:10.2140/apde.2012.5.705

Mathematical Reviews number (MathSciNet)
MR3006640

Zentralblatt MATH identifier
1264.35215

#### Citation

Ionescu, Alexandru; Pausader, Benoit; Staffilani, Gigliola. On the global well-posedness of energy-critical Schrödinger equations in curved spaces. Anal. PDE 5 (2012), no. 4, 705--746. doi:10.2140/apde.2012.5.705. https://projecteuclid.org/euclid.apde/1513731243

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