Helein F. Harmonic Maps, Conservation Laws and Moving Frames

Издательство Cambridge University Press, 2002, -290 pp.Harmonic maps between Riemannian manifolds provide a rich display of both differential geometric and analytic phenomena. These aspects are inextricably intertwined — a source of undiminishing fascination.
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.
Now let us try to imagine a 3-dimensional matter with analogous properties. We can stretch it inside any geometrical manifold, as for instance a sphere: although our 3-dimensional body will be confined — since generically lines will shrink to points — it may find an equilibrium configuration. Actually the mathematical description of such a situation, which is apparently more abstract than the previous ones, looks like the mathematical description of a nematic liquid crystal in equilibrium. Such a bulk is made of thin rod shaped molecules (nema means thread in Greek) which try to be parallel each to each other. Physicists have proposed different models for these liquid crystals where the mean orientation of molecules around a point in space is represented by a vector of norm 1(hence some point on the sphere). Thus we can describe the configuration of the material using a map defined in the domain filled by the liquid crystal, with values into the sphere. We get a situation which is mathematically analogous to the abstract experiment described above, by imagining we are trying to imprison a piece of perfectly elastic matter inside the surface of a sphere. The physicists Oseen and Frank proposed a functional on the set of maps from the domain filled with the material into the sphere, which is very close to the elastic energy of the abstract ideal matter.Geometric and analytic setting
Harmonic maps with symmetry
Compensations and exotic function spaces
Harmonic maps without symmetry
Surfaces with mean curvature in L²