Wave map Cauchy data

Initial position ${\varphi }_0$ for a wave map takes values in the target, and initial velocity ${\varphi }_1$ belongs to its tangent space at ${\varphi }_0(x)$. For $S^2$ this means $|{\varphi }_0|=1$ and ${\varphi }_0\cdot {\varphi }_1=0$. Localized data agree with a constant map and zero velocity outside a compact set; the map itself need not be a compactly supported ambient function.