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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.

The wave map scaling symmetry is ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$. Its conserved wave map energy isThus energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The scaling-critical homogeneous Sobolev spaces for perturbations of a constant map are ${\dot{H}}^{n/2}\times {\dot{H}}^{n/2-1}$. These relations constitute wave map energy and criticality.

A time-independent wave map is a harmonic map of its spatial domain into the target. For target $S^2$, its equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. In two dimensions inverse stereographic projection gives a smooth nonconstant example of finite wave map energy $4\pi$.

Under ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$, the conserved wave map energy scales as ${\lambda }^{n-2}E$. Energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The critical derivative index of the homogeneous Sobolev space is $s_c=n/2$. Small energy alone is not an appropriate general small-data regularity hypothesis in supercritical dimensions.