Conformally Invariant Variational Problems. - SAM

plane, moreover we combine (X.246) and (X.249) and we obtain
that (X.249) is equivalent to
⎧
I(iA) = 0
⎪⎨ ( )
⃗e 1 ∧I ⃗V +2iW ⃗ = 0 (X.250)
( [ ])
⎪⎩ ⃗e 2 ∧I i ⃗V +2iW ⃗ = 0
Combining(X.244)and (X.250)we obtainthat the conservation
laws (X.239) are equivalent to
⎧
⎨ A = 0
⎩ ⃗V = −2iW ⃗ (X.251)
= −2iπ ⃗n (∂ zH)
⃗
Or in other words, for a conformal immersion ⃗ Φ of the disc into
R m , there exists ⃗ L from D 2 into R m such that (X.239) holds if
and only if there exists a complex valued function B and a map
⃗L from D 2 into R m such that
∂ z
⃗ L = B⃗ez −2iπ ⃗n (∂ z
⃗ H) .
(X.252)
We shall now exploit the crucial fact that ⃗ L is real valued by
taking ∂ z of (X.252). Let
f := e λ B +2ie 2λ 〈 ⃗ H, ⃗ H0
〉
. (X.253)
With this notation (X.252) becomes 79
〈 〉
∂ zL ⃗ = e −λ f⃗e z −2i ⃗H, H0 ⃗ ∂ zΦ−2iπ⃗n ⃗ (∂ zH) ⃗ . (X.255)
79 In real notations this reads also, after using the identity (X.101) in lemma X.3
⎛ ⎞ ⎛ ⎞
a b ∂ x2Φ
⃗
∇ ⊥ L ⃗ = e
−2λ⎝
⎠ ⎜ ⎟
⎝ ⎠+∇H ⃗ −3π ⃗n (∇H)+⋆∇ ⃗ ⊥ ⃗n∧ H ⃗ (X.254)
−b a ∂ x1Φ ⃗
where f = a+ib.
202