Global well-posedness and scattering for the defocusing, energy -critical, nonlinear Schr{\"o}dinger equation in the exterior of a convex obstacle when $d=4$

In this paper we prove that the energy - critical nonlinear Schr{"o}dinger equation in the domain exterior to a convex obstacle is globally well - posed and scattering for initial data having finite energy. To prove this we utilize frequency localized Morawetz estimates adapted to an exterior domain.