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For smooth localized compatible wave map Cauchy data and compact Riemannian target, one-dimensional wave maps stay smooth globally. In null coordinates, the equation is $D_v{\varphi }_u=D_u{\varphi }_v=0$. Compatibility of the target metric with the covariant derivative makes $|{\varphi }_u{|}^2$ independent of $v$ and $|{\varphi }_v{|}^2$ independent of $u$. These transported derivative bounds and higher energy estimates prevent finite-time breakdown.

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.

Smooth compatible wave map Cauchy data give a unique local smooth wave map. Relative to a constant map, Sobolev spaces $H^s\times H^{s-1}$ with $s>n/2+1$ provide a classical local theory. Iteration for the semilinear wave equation, Sobolev algebra and energy estimates give existence, uniqueness and continuous dependence. The smooth continuation criterion for semilinear wave equations extends the solution while these norms stay bounded. The sphere constraint and tangency constraint remain satisfied.

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.

Global regularity for one-dimensional wave maps holds for arbitrary smooth localized compatible data. In retarded and advanced null coordinates $u=t-x$, $v=t+x$, the wave map equation says $D_v{\varphi }_u=D_u{\varphi }_v=0$, with $D$ the target covariant derivative. Hence $|{\varphi }_u{|}^2$ depends only on $u$, and $|{\varphi }_v{|}^2$ only on $v$. Initial derivative bounds persist. Compactness of the sphere and differentiated energy estimates then prevent finite-time loss of smoothness.

Smooth compatible data sufficiently small in high weighted Sobolev norms relative to a constant map produce global smooth wave maps in three and four spatial dimensions. In four dimensions, derivative decay $(1+t{)}^{-3/2}$ is time-integrable and closes commuted wave energy estimates. In three dimensions the weaker $(1+t{)}^{-1}$ decay requires the cancellation of null forms for wave equations, exploited by the vector field method for wave equations. These classical localized-data statements do not assert global regularity from small supercritical energy alone.

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 $4\pi$.

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.

The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.