Conformally Invariant Variational Problems. - SAM

This last identity written with the complex coordinate z is exactly
(X.100) and lemma X.3 is proved. ✷
Before to move to the proof of theorem X.7 we shall need two
more lemma. First we have
Lemma X.4. 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.111)
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 )]
Then the following identities hold
and
∂ z
[
e λ ⃗e z
]
=
e 2λ
2 ⃗ H , (X.112)
∂ z
[
e −λ ⃗e z
]
=
1
2 ⃗ H 0 . (X.113)
✷
Proof of lemma X.4. The first identity (X.112) comes simply
from the fact that ∂ z ∂ zΦ ⃗ =
1
4 ∆⃗ Φ, from (X.117) and the expression
of the mean curvature vector in conformal coordinates that
we have seen several times and which is given by
⃗H = e−2λ
2 ∆⃗ Φ .
It remains to prove the identity (X.113). One has moreover
[ ]
∂ z e λ ⃗e z = ∂z ∂ zΦ ⃗
1
[
]
= ∂ 2 x
⃗Φ−∂ 2
4
2 1 x
⃗Φ−2i ∂ 2 ⃗ 2
2
x 1 x 2Φ . (X.114)
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