The contemplation of the atlas of an airline company always offers us something puzzling: the trajectories of the airplanes look curved, which goes against our basic intuition, according to which the shortest path is a straight line. One of the reasons for this paradox is nothing but a simple geometrical fact: on the one hand our earth is round and on the other hand the shortest path on a sphere is an arc of great circle: a curve whose projection on a geographical map rarely coincides with a straight line. Actually, choosing the trajectories of airplanes is a simple illustration of a classical variational problem in differential geometry: finding the geodesic curves on a surface, namely paths on this surface with minimal lengths.

Using water and soap we can experiment an analogous situation, but where the former path is now replaced by a soap film, and for the surface of the earth — which was the ambient space for the above example — we substitute our 3-dimensional space. Indeed we can think of the soap film as an excellent approximation of some ideal elastic matter, infinitely extensible, and whose equilibrium position (the one with lowest energy) would be either to shrink to one point or to cover the least area. Thus such a film adopts a minimizing position: it does not minimize the length but the area of the surface. Here is another classical variational problem, the study of minimal surfaces.

In the previous chapter we saw some regularity results for weak solutions of non-linear partial differential equations, whose proofs are based on the fact that, among the quadratic combinations of derivatives of functions, the Jacobian determinants have a subtle extra regularity. To our knowledge, this phenomenon was first used by Henry Wente [177]. It is analogous to the compensations that allow one to pass to the limit in certain non-linear combinations of weakly convergent sequences, and which were used by François Murat and Luc Tartar in their compensated compactness theory [122], [166].

Below, we follow the clear presentation, due to Haїm Brezis and Jean-Michel Coron [21], of H. Wente's discovery, and we also present some of its improvements. In particular, twenty years after H. Wente's result, works by Stefan Müller and also by Ronald Coifman, Pierre-Louis Lions, Yves Meyer and Stephen Semmes resulted in a finer study of these compensation phenomena, in particular using Hardy spaces.

Other function spaces, less known to specialists in partial differential equations, such as Lorentz spaces, have also proven to be useful. These developments are presented in sections 3.2 (Hardy spaces) and 3.3 (Lorentz spaces) of this chapter. As an example of the use of Hardy spaces we present the regularity theorem for weakly stationary maps, due to Lawrence C. Evans, in the last section of this chapter. Most of the analytic results of this chapter will be applied in chapters 4 and 5, to other geometrical settings.

This chapter essentially describes the objects and properties that will interest us in this work. For a more detailed exposition of the general background in Riemannian geometry and in analysis on manifolds, one may refer for instance to [183] and [98]. After recalling how to associate, to each Riemannian metric on a manifold, a Laplacian operator on the same manifold, we will give a definition of smooth harmonic map between two manifolds. Very soon, we will use the variational framework, which consists in viewing harmonic maps as the critical points of the Dirichlet functional.

Next, we introduce a frequently used ingredient in this book: Noether's theorem. We present two versions of it: one related to the symmetries of the image manifold, and the other which is a consequence of an invariance of the problem under diffeomorphisms of the domain manifold (in this case it is not exactly Noether's theorem, but a “covariant” version).

These concepts may be extended to contexts where the map between the two manifolds is less regular. In fact, a relatively convenient space is that of maps with finite energy (Dirichlet integral), H1 (M, N). This space appears naturally when we try to use variational methods to construct harmonic maps, for instance the minimization of the Dirichlet integral. The price to pay is that when the domain manifold has dimension larger than or equal to 2, maps in H1 (M, N) are not smooth, in general. Moreover, H1 (M, N) does not have a differentiable manifold structure.

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.