On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces

TL;DR

This paper develops a method to establish global well-posedness and scattering for energy-critical Schrödinger equations on curved noncompact manifolds, specifically hyperbolic space, by adapting concentration compactness and Morawetz inequalities to the geometry.

Contribution

It extends the Kenig-Merle method to variable coefficient settings on curved spaces, proving global results for energy-critical Schrödinger equations on hyperbolic space.

Findings

Proved global well-posedness and scattering in H^1 for the energy-critical defocusing Schrödinger equation on H^3.

Adapted concentration compactness and Morawetz inequalities to curved geometries.

Extended Euclidean techniques to noncompact Riemannian manifolds.

Abstract

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger 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-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+\Delta_\g)u=u|u|^{4} on the hyperbolic space $H^3$.