Improved Global Well-Posedness for Mass-Critical Nonlinear Schrödinger Equations on Tori

We show new global well-posedness results for mass-critical nonlinear Schrödinger equations on tori in one and two dimensions. For the quintic nonlinear Schrödinger equation on the circle we show global well-posedness for initial data in Hs(T) for s>13 and ‖u0‖L2(T)≪1. In two dimensions we show global well-posedness on possibly irrational tori for s>35. In the focusing case we need to consider small L2-norm, whereas in the defocusing case large mass is covered.