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We consider the focusing energy-critical nonlinear Schr\"odinger equation$iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove thatif a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\inI}\|\nabla u(t)\|_2<\|\nabla W\|_2$, then it is global and scatters bothforward and backward in time. Here $W$ denotes the ground state, which is astationary solution of the equation. In particular, if a solution has bothenergy and kinetic energy less than those of the ground state $W$ at some pointin time, then the solution is global and scatters. We also show that anysolution that blows up with bounded kinetic energy must concentrate at leastthe kinetic energy of the ground state. Similar results were obtained by Kenigand Merle in \cite{Evian, kenig-merle} for spherically symmetric initial dataand dimensions $d=3,4,5$.

The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher