Conformally Invariant Variational Problems. - SAM

In one hand the projection into the normal direction gives
[
]
π ⃗n ∂ 2 x
⃗Φ−∂ 2 2
1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ = 2e 2λ H0 ⃗ . (X.115)
In the other hand the projection into the tangent plane gives
[
]
π T ∂ 2 x
⃗Φ−∂ 2 2
1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ 〈 [
]〉
= e −λ ∂ x1Φ, ⃗ ∂ 2 x
⃗Φ−∂ 2 2
1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ 〈 [
]〉
+e −λ ∂ x2Φ, ⃗ ∂ 2 x
⃗Φ−∂ 2 2
1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ This implies after some computation
[
]
π T ∂ 2 x
⃗Φ−∂ 2 2
1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ ⃗e 1
⃗e 2 .
= 2e λ [∂ x1 λ−i∂ x2 λ] ⃗e 1 −2e λ [∂ x2 λ+i∂ x1 λ] ⃗e 2
(X.116)
= 8 ∂ z e λ ⃗e z .
The combination of (X.114), (X.115) and (X.116) gives
which implies (X.113).
∂ z
[
e λ ⃗e z
]
=
e 2λ
2 ⃗ H 0 +2∂ z e λ ⃗e z ,
The last lemma we shall need in order to prove theorem X.7
is the Codazzi-Mainardiidentitythat we recall and prove below.
Lemma X.5. [Codazzi-Mainardi Identity.] Let ⃗ Φ be a conformal
immersion of the disc D 2 into R m , denote z := x 1 +ix 2 ,
e λ := |∂ x1
⃗ Φ| = |∂x2 ⃗ Φ| denote
⃗e i := e −λ ∂ xi
⃗ Φ ,
✷
(X.117)
and let ⃗ H 0 be the Weingarten Operator of the immersion expressed
in the conformal coordinates (x 1 ,x 2 ) :
⃗H 0 := 1 2 [I(⃗e 1,⃗e 1 )−I(⃗e 1 ,⃗e 1 )−2iI(⃗e 1 ,⃗e 2 )]
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