 Solution

Compatible Cauchy data for a wave map are ${\varphi }_0:{\mathbb{R}}^n\to S^2$ and ${\varphi }_1(x)\in T_{{\varphi }_0(x)}{S}^2$: thus $|{\varphi }_0|=1$ and ${\varphi }_0\cdot {\varphi }_1=0$. Smooth localized wave map Cauchy data equal a constant $p\in S^2$ with zero velocity outside a compact set. It is ${\varphi }_0-p$, rather than the sphere-valued map itself, that has compact support. The geometric constraints propagate under the wave map equation.