The quintic NLS on perturbations of ${\mathbf{R}}^3$

Abstract

Consider the defocusing quintic nonlinear Schrödinger equation on ${\mathbf{R}}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.

Added an appendix detailing the L^1->L^\infty estimate previously mentioned in passing; minor expository improvements