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A harmonic map between Riemannian manifolds is a critical point of the Dirichlet energy $\frac{1}{2}\int |dQ{|}^2$. For a Euclidean domain and target unit sphere, the equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. Stationary wave maps are exactly harmonic maps of their spatial domain.

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.