Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on ${\text{T}}^3$ with intial data lying in its critical space $H^{\frac{1}{2}}({\text{T}}^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^{\mathrm{\infty }}{H}^{\frac{1}{2}}(I\times {\text{T}}^3)$, then $u$ is global.