The authors prove that solutions of the three‑dimensional focusing energy‑critical nonlinear Schrödinger equation exist for all time and disperse, provided the initial data lie below the natural ground‑state thresholds. In plain terms, they show that if the initial wave is smaller than a special steady solution both in energy and in mass, then the wave never collapses and eventually behaves like a free solution. This confirms a long‑standing threshold conjecture in three dimensions.

The equation studied is the focusing energy‑critical NLS in three space dimensions, the model with a quintic (fifth‑power) nonlinearity that preserves the same scaling as the energy. A special stationary solution called the ground state W sets the sharp threshold. The result assumes the initial data u0 lie in the energy space H^1(R^3) and satisfy E(u0)<E(W) and L^2 mass smaller than that of W. Under these hypotheses the solution is global in time, belongs to the usual spacetime control class, and scatters in both time directions to linear solutions.

Technically the proof overcomes a well known obstacle in the nonradial case: poor control of low frequencies. The authors build a frequency‑localized interaction Morawetz identity. This is an integral identity that pairs pieces of the solution at different frequencies and produces a positive quartic spacetime term — an L^4 in time and space bound — even in the focusing setting. They adapt the interaction weight to a projected density (a truncated low‑frequency part) so the principal term is coercive below the ground state. Nonlinear cancellations then let them absorb the errors caused by frequency truncation into this positive term.

The new L^4 spacetime estimate is fed into the usual concentration–compactness strategy. If scattering failed, one obtains a minimal “critical’’ solution that is almost periodic in time. The quartic estimate rules out such a nonzero critical solution, even without assuming finite mass. That contradiction completes the proof that all allowed initial data must scatter. The argument refines and extends techniques developed for related defocusing and radial problems.