Conformally Invariant Variational Problems. - SAM

Then the following identity holds
e −2λ ∂ z
(e 2λ < H, ⃗ H ⃗ )
0 > =< H,∂ ⃗ zH ⃗ > + < H0 ⃗ ,∂ zH ⃗ > .
Thus
Proof of lemma X.5. Using (X.113) we obtain
< ∂ zH0 ⃗ , H ⃗ )]
>= 2
〈∂ z
[∂ z
(e −2λ ∂ zΦ ⃗ , H ⃗ 〉
)]
= 2
〈∂ z
[∂ z
(e −2λ ∂ zΦ ⃗ , H ⃗ 〉
.
(X.118)
✷
< ∂ z
⃗ H0 , ⃗ H >
]
= −4
〈∂ z
[∂ z λ e −2λ ∂ zΦ ⃗ , H ⃗ 〉
+
〈
= −2∂ z λ ⃗H0 , H〉
⃗ 〉
+
〈∂ zH, ⃗ H ⃗
〈∂ z
[ e
−2λ
.
2
] 〉
∆Φ
⃗ , H ⃗
This last identityimpliesthe Codazzi-Mainardiidentity(X.118)
and lemma X.5 is proved.
✷
Proof of theorem X.7. Du to lemma X.3, as explained in
remark X.2, it suffices to prove in conformal parametrization
the identity (X.103). First of all we observe that
]))
4e −2λ R
(π ⃗n
(∂ z
[π ⃗n (∂ zH) ⃗
[
= e −2λ π ⃗n
(div π ⃗n (∇H)
⃗ ])
= ∆ ⊥
⃗ H
(X.119)
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