On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2

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For the focusing mass-critical NLS \(iu_t + \Delta u = - \left| u \right|^{\tfrac{4} {d}} u\), it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for Hx1 initial data in dimensions d = 2, 3 with spherical symmetry and d ⩾ 4 with certain splitting-spherically symmetric initial data.