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 H 1 for the energy-critical defocus-on hyperbolic space ‫ވ‬ 3 .

Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case

“…For the proof, we refer to Bray [5] for (2.11), Ionescu, Pausader, Staffilani [18] for (2.12), Hebey [16] for (2.13), see also Lawrie, Oh, Shahshahani [33]. (2.14) is obtained in [50].…”

“…The intrinsic and extrinsic formulations are equivalent in the following sense, see [Section 2, [37]]. 18) in the sense that there exist continuous functions P, Q such that…”

The energy-critical NLS with inverse-square potential

“…This paper is not the first to implement the credo that to treat critical dispersive equations with broken symmetries, one must first solve the limiting problems where the symmetries are restored. Previous examples include [15,16,17,18,19,24,26,27,31,36]. This is the first model to be studied, however, which retains scale invariance, but loses space translation invariance.…”

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

“…, was established similar to [12,13]. To show the space-time control of the nonlinear profiles associated with the profile e −it k ∆ R×T 3 , this loss of 2 3 derivatives does not allow the local wellposedness theory in H 1 .…”