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We construct a center-stable manifold of the ground state solitons in theenergy space for the critical wave equation without imposing any symmetry, asthe dynamical threshold between scattering and blow-up, and also as acollection of solutions which stay close to the ground states. Up to energyslightly above the ground state, this completes the 9-set classification of theglobal dynamics in our previous paper. We can also extend the manifold toarbitrary energy size by adding large radiation. The manifold contains all thesolutions scattering to the ground state solitons, and also some of thoseblowing up in finite time by concentration of the ground states.

Center-stable manifold of the ground state in the energy space for the critical wave equation