We come back to the study of harmonic maps, but now we drop the symmetry hypothesis on the image manifold. It is then clear that most of the methods seen in chapter 3 are no longer valid. Nevertheless, it is tempting to try to adapt these results to our new situation. This is the naïve point of view we will adopt in this chapter.

The tool we will use the most, and which will replace the conservation laws in chapter 3, is an orthonormal moving frame on the image manifold: it turns out that this choice of representation, used a century ago by Gaston Darboux for the study of surfaces, and developed by Elie Cartan, is very efficient for studying harmonic maps. We remark that, as with all geometric coordinate systems, there is not only one, but infinitely many ways of defining orthonormal tangent frame fields. Instead of being an inconvenience, this abundance of choice is an advantage since one passes from one orthonormal frame field to another through the action of a gauge group.

In this way, symmetries re-enter, and Noether's theorem is not far away: by choosing a “Coulomb frame”, the orthonormal frame selected satisfies an equation which may be written as a conservation law.

The use of “Coulomb frames” is fundamental for the regularity theorems for harmonic maps which will be presented in the first three sections. We will also need exotic function spaces (Hardy, BMO, Lorentz) and some results from the previous chapter.