Global well-posedness for the energy-critical focusing nonlinear Schrodinger equation on T4

In this paper, we consider the global (in time) well-posedness for the focusing cubic nonlinear Schrodinger equation (NLS) on 4-dimensional tori -either rational or irrational- and with initial data in H-1. We prove that if a maximal-lifespan solution of the focusing cubic NLS u : I x T-4 -> C satisfies suPp(t is an element of I) parallel to u(t)parallel to((H)over dot1(T4)) < parallel to W parallel to((H)over dot1(R4)), (R4), 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 R-4 . As a consequence, we also construct the global solution with some threshold conditions related to the modified energy of the initial data which is the energy modified by the mass of the initial data and the best constants of Sobolev embedding on T-4. (C) 2021 Published by Elsevier Inc.