The author considers ‘wave maps’ \(U:M\to S^3\) where \(M\) is Minkowski space and \(S^3\) is the unit three-sphere. By definition, these are critical for \(\int_M|\nabla U|^2\) where \(|\quad|^2\) is the Hilbert-Schmidt form determined by the Lorentzian metric on \(M\) and the Riemannian metric on \(S^3\). Such mappings are often called sigma models. The analogous mappings between Riemannian manifolds are called harmonic.
An example of such a wave mapping was found by J. Shatah [Commun. Pure Appl. Math. 41, 459-469 (1988; Zbl 0686.35081)]. It is invariant under \({\text{SO}}(3)\)-rotations and also dilations. The way that it blows up seems typical, in numerical studies of the evolution of initial data under the wave map equations.
The author finds the general wave map \(M\to S^3\) invariant under rotations and dilations in this way. With these symmetries, the wave mapping equations reduce to an ordinary but nonlinear differential equation. He finds that Shatah’s example is the first in a series of solutions, which may be regarded as excitations of this first example: they may be labelled by a degree that counts the number of oscillations. It is a natural and interesting set of examples.