Conformally Invariant Variational Problems. - SAM

either ∂ z u = 1 2 (∂ x 1
u−i∂ x2 u) ≡ 0
(II.7)
or ∂¯z u = 1 2 (∂ x 1
u+i x2 u) ≡ 0
(II.8)
✷
Proof of Proposition II.1
Because of (II.1) a map u from a 2-dimensional domain U of
C into a manifold (N n ,h) is conformal if and only if one has
⎧
⎨ < ∂ x1 u,∂ x2 u > h = 0
(II.9)
⎩
| ∂ x1 u | h =| ∂ x2 u | h
in the case of U : U CC −→ C a direct computation gives
∂u
∂z
∂ū
∂z = 1 4 [(∂ x 1
u 1 +∂ x2 u 2 )+i(∂ x1 u 2 −∂ x2 u 1 )]
[(∂ x1 u 1 −∂ x2 u 2 )−i(∂ x2 u 1 +∂ x1 u 2 )]
(II.10)
= 1 4
[
| ∂x1 u | 2 − | ∂ x2 u | 2 −2i < ∂ x1 u,∂ x2 u > ]
where < ·,· > denotes the scalar product in C. Observe that
the quadratic 1−0⊗1−0 form Φ(u) given by
Φ(u) = ∂u
∂z
∂ū
∂z
dz ⊗dz
is invariant under holomorphic change of coordinates:
for w(z) satisfying ∂¯z w = 0 one has locally away from zeros
of w ′
φ(u) = ∂u
∂w
∂ū
∂w dw⊗dw
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