Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 10 / 4 / 3 / Solution

Harmonic maps give nonconstant stationary wave maps. For $n=2$, inverse stereographic projection givesIt satisfies $|Q|=1$ and $\Delta Q=-|\nabla Q{|}^{2}Q$, so $\varphi (t,x)=Q(x)$ is a smooth global wave map. Its wave map energy is $4\pi$, since $|\nabla Q{|}^2=8/(1+|x{|}^2{)}^2$. Translations and rescalings give further examples; these harmonic maps approach a constant at infinity.