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

For the focusing mass-critical NLS $iu_t+\Delta u=-|u|^{\frac 4d}u$, 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 $H^1_x$ initial data in dimensions $d=2,3$ with spherical symmetry and $d\ge 4$ with certain splitting-spherically symmetric initial data.