Global well-posedness of the energy-critical nonlinear Schr\"odinger equations on $\mathbb{T}^{d}$

In this paper, we prove the global well-posedness of the energy-critical nonlinear Schr\"odinger equations on the torus $\mathbb{T}^{d}$ for general dimensions. This result is new for dimensions $d\ge5$, extending previous results for $d=3,4$ [10,22]. Compared to the cases $d=3,4$, the regularity theory for higher $d$, developed in the underlying local well-posedness result [17], is less understood. In particular, stability theory and inverse inequalities, which are ingredients in [10,22] and more generally in the widely used concentration compactness framework since [13], are too weak to be applied to higher dimensions. Our proof introduces a new strategy for addressing global well-posedness problems. Without relying on perturbation theory, we develop tools to analyze the concentration dynamics of the nonlinear flow. On the way, we show the formation of a nontrivial concentration.