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We consider the minimal mass $m_0$ required for solutions to themass-critical nonlinear Schr\"odinger (NLS) equation $iu_t + \Delta u = \mu|u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists aminimal-mass solution blowing up (in the sense of an infinite spacetime norm)in both time directions, whose orbit in $L^2_x(\R^d)$ is compact afterquotienting out by the symmetries of the equation. A similar result is obtainedfor spherically symmetric solutions. Similar results were previously obtainedby Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas,\cite{begout}, in dimensions $d\geq 3$ for the mass-critical NLS and by Kenigand Merle, \cite{merlekenig}, in the energy-critical case. In a subsequentpaper we shall use this compactness result to establish global existence andscattering in $L^2_x(\R^d)$ for the defocusing NLS in three and higherdimensions with spherically symmetric data.

Minimal-mass blowup solutions of the mass-critical NLS