Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS

Let d ≥ 4 and let u be a global solution to the focusing masscritical nonlinear Schrödinger equation iut + ∆u = −|u| 4 d u with spherically symmetric H x initial data and mass equal to that of the ground state Q. We prove that if u does not scatter then, up to phase rotation and scaling, u is the solitary wave eQ. Combining this result with that of Merle [15], we obtain that in dimensions d ≥ 4, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.