Conformally Invariant Variational Problems. - SAM

frames, thereby compensating for the lack of global symmetry
with “infinitesimal symmetries”.
This method, although seemingly unnatural and rather mysterious,hassubsequentlyprovedveryefficienttoanswerregularity
and compactness questions, such as in the study of nonlinear
wave maps (see [FMS], [ShS], [Tao1], [Tao2]). For this reason,
it is worthwhile to dwell a bit more on Hélein’s method.
We first recall the main result of F. Hélein.
Theorem VII.2. [He] Let N n be a closed C 2 -submanifold of
R m . Suppose that u is a harmonic map in W 1,2 (D 2 ,N n ) that
weakly satisfies the harmonic map equation (VI.26). Then u
lies in C 1,α for all α < 1.
Proof of theorem VII.2 when N n is a two-torus.
The notion of harmonic coordinates has been introduced first
in general relativity by Yvonne Choquet-Bruaht in the early
fifties. She discovered that the formulation of Einstein equation
in these coordinates simplifies in a spectacular way. This
idea of searchingoptimalchartsamongallpossible ”gauges”has
also been very efficient for harmonic maps into manifolds. Since
the different works of Hildebrandt, Karcher, Kaul, Jäger, Jost,
Widman...etc in the seventies it was known that the intrinsic
harmonic map system (VI.18) becomes for instance almost ”triangular”
in harmonic coordinates (x α ) α in the target which are
minimizing the Dirichlet energy ∫ U |dxα | 2 g dvol g . The drawback
of this approach is that working with harmonic coordinates requirestolocalizeinthetargetandtorestrictonlytomapstaking
valuesintoasingle chartinwhichsuchcoordinatesexist! While
looking at regularity question this assumption is very restrictive
as long as we don’t know that the harmonicmap u is continuous
for instance. It is not excluded a priori that the weak harmonic
map u we are considering ”covers the whole target” even locally
in the domain.
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