Finite-mass soliton-type rigidity and four-channel reduction for the three-dimensional nonradial focusing energy-critical nonlinear Schr\"odinger equation

Abstract

The concentration--compactness channels below the ground-state threshold are investigated for the three-dimensional nonradial focusing energy-critical nonlinear Schr\"odinger equation. After one-sided normalization, a minimal critical element falls into four classes: the finite-time, rapid-cascade, bounded-scale finite-mass, and residual quasi-soliton channels. The first three classes are rigorously excluded. The main result shows, without radial symmetry, zero momentum, or a slow spatial center, that every finite-mass bounded-scale almost-periodic solution is identically zero. Consequently, any minimal counterexample to below-threshold scattering must lie in the residual quasi-soliton channel; if its scale is bounded, then it has infinite mass at every time.