In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational or irrational - and with initial data in $H^1$. Furthermore, we prove that if a maximal-lifespan solution of the focusing cubic NLS $u: I\times\mathbb{ T}^4\to \mathbb{ C}$ satisfies $\sup_{ t\in I}\|u(t)\|_{ \dot{ H}^1(\mathbb{ T}^4)}<\|W\|_{ \dot{ H}^1(\mathbb{ R}^4)}$, then it is a global solution. $W$ denotes the ground state on Euclidean space, which is a stationary solution of the corresponding focusing equation in $\mathbb{ R}^4$.
Categories: math.AP