Global Well-posedness and Scattering for the Defocusing Cubic nonlinear Schrödinger equation in Four Dimensions

In this short note, we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical nonlinear Schrödinger equation (NLS) in four space dimensions obtained previously by Ryckman and Visan [“Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in $\mathbb R^{1+4}$.” American Journal of Mathematics 129 (2007): 1–60. MR2288737]. The argument is inspired by the recent work of Dodson [“Global well-posedness and scattering for the defocusing, L²-critical, nonlinear Schrödinger equation when d≥3.” (2009): preprint arXiv:0912.2467.] on the mass-critical NLS.