Characterization of large energy solutions of the equivariant wave map problem: II

We consider $1$-equivariant wave maps from $\Bbb{R}^{1+2}\to\Bbb{S}^2$ of finite energy. We establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map~$Q$. In particular, for each global energy solution of topological degree one, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with a companion article (Part I), where we consider the case of finite-time blow up, this gives a characterization of all $1$-equivariant, degree~$1$ wave maps in the energy regime $[E(Q),3E(Q))$.