Abstract

abstract:

Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on ${\bf R}^3$ 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.