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We consider the minimal mass  m  0  required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation  iu t   + Δ u  = μ| u | 4/ d   u  to blow up. If  m  0  is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in    is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal.  235  (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the  L  2 -critical nonlinear Schrödinger equation, preprint], in dimensions  d  ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in    for the defocusing NLS in three and higher dimensions with spherically symmetric data.

Minimal-mass blowup solutions of the mass-critical NLS