Conformally Invariant Variational Problems. - SAM

The main idea of Frederic Hélein was to extend the notion
of harmonic coordinates of Choquet-Bruhat by replacing it with
the more flexible harmonic or Coulomb orthonormal frame, notion
for which no localization in the target is needed anymore.
Precisely the Coulomb orthonormal frames are mappings e =
(e 1 ,··· ,e n ) from the domain D 2 into the orthonormal basis
of T u N n minimizing the Dirichlet energy but for the covariant
derivatives D g in the target :
∫
D 2
n∑
∫
|D g e i | 2 dxdy =
i=1
D 2
n∑
|(e k ,∇e i )| 2 dxdy .
i,k=1
where (·,·) denotes the canonical scalar product in R m .
We will consider the case when N n is a two-dimensional parallelizable
manifold (i.e. admitting a global basis of tangent
vectors for the tangent space), namely a torus T 2 arbitrarilyimmersed
into Euclidean space R m , for m large enough. The case
of the two-torus is distinguished. Indeed, in general, if a harmonic
map u takes its values in an immersed manifold N n , then
it is possible to lift u to a harmonic map ũ taking values in a
parallelizable torus (S 1 ) q of higher dimension. Accordingly, the
argument which we present below can be analogously extended
to a more general setting 15 .
Let u ∈ W 1,2 (D 2 ,T 2 ) satisfy weakly (VI.26). We equip T 2
with a global, regular, positive orthonormal tangent frame field
(ε 1 ,ε 2 ). Let ẽ := (ẽ 1 ,ẽ 2 ) ∈ W 1,2 (D 2 ,R m × R m ) be defined by
the composition
ẽ i (x,y) := ε i (u(x,y)) .
The map (ẽ) is defined on D 2 and it takes its values in the
15 although the lifting procedure is rather technical. The details are presented in Lemma
4.1.2 from [He].
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