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 $H^s(\mathbb{T})$ for $s>\frac{1}{3}$ and $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2(\mathbb{T})}\ll 1$. In two dimensions we show global well-posedness on possibly irrational tori for $s>\frac{3}{5}$. In the focusing case we need a smallness condition for the mass, whereas in the defocusing case large mass is covered.