Conformally Invariant Variational Problems. - SAM

This is the main advantage of working with E instead of L.
Similarly, in two dimension, the energy functional E shares the
same advantage over the area functional A : while A is invariant
under the action of the infinite group of diffeomorphisms of D 2
into itself, the Dirichlet energy is invariant under the group of
positiveconformaldiffeomorphismgroupofthedisc, the Möbius
group M + (D 2 ) which is 3 dimensionalas we saw in the previous
subsection and given by the holomorphic maps of the form
for some θ ∈ R and a ∈ D 2 .
iθ w −a
f(w) := e
1−aw
This invariance of E under conformal transformations may
easilybe seenbyworkingwiththecomplexvariablez = x 1 +ix 2 .
Indeed, we note
et
∂ z := 1 2 (∂ x 1
−i∂ x2 )
∂ z := 1 2 (∂ x 1
+i∂ x2 )
so that du = ∂ z udz +∂ z udz, and thus
E(u) = i ∫
|∂ z u| 2 +|∂ z u| 2 dz ∧dz .
2 D 2
Accordingly, if we compose u with a conformal transformation,
i.e. holomorphic, z = f(w), there holds for ũ(w) = u(z) the
identities
|∂ w ũ| 2 = |f ′ (w)| 2 |∂ z u| 2 ◦f and |∂ w ũ| 2 = |f ′ (w)| 2 |∂ z u| 2 ◦f .
Moreover, dz∧dz = |f ′ (w)| 2 dw∧dw. Bringing altogetherthese
results yields the desired conformal invariance E(u) = E(ũ).
An heuristic argument shed some light on the reason to believe
that the strategy of minimizing the energy E should provide
a minimizer of A. Assuming one moment that we would
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