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

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classication of the global dynamics in our previous paper [14]. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in nite time by concentration of the ground states.