Conformally Invariant Variational Problems. - SAM

Proof of lemma X.3. We denote by (⃗e 1 ,⃗e 2 ) the orthonormal
basis of ⃗ Φ ∗ (TD 2 ) given by
⃗e i = e −λ ∂⃗ Φ
∂x i
.
With these notations the second fundamental form h which is a
symmetric 2-form on TD 2 into ( ⃗ Φ ∗ TD 2 ) ⊥ is given by
h = ∑ α,i,j hα ij ⃗n α ⊗(⃗e i ) ∗ ⊗(⃗e j ) ∗
with h α ij = −e−λ (
∂⃗nα
∂x i
,⃗e j
)
(X.104)
We shall also denote
⃗ hij := ⃗ I(⃗e i ,⃗e j ) =
m−2
∑
α=1
h α ij ⃗n α
In particular the mean curvature vector ⃗ H is given by
⃗H =
m−2
∑
α=1
H α ⃗n α = 1 m−2
∑
(h α 11 +h α
2
22)⃗n α = 1 2 (⃗ h 11 + ⃗ h 22 ) (X.105)
α=1
Let ⃗n be the m − 2 vector of R m given by ⃗n = ⃗n 1 ∧ ··· ∧⃗n 2 .
We identify vectors and m − 1-vectors in R m using the Hodge
operator ⋆ of R m for the canonical flat metric. Hence we have
for instance
⋆(⃗n∧⃗e 1 ) =⃗e 2 and ⋆(⃗n∧⃗e 2 ) = −⃗e 1 (X.106)
Since⃗e 1 ,⃗e 2 ,⃗n 1···⃗n m−2 is a basis of T ⃗Φ(x1 ,x 2 ) Rm , we can write for
every α = 1···m−2
∇⃗n α =
m−2
∑
β=1
< ∇⃗n α ,⃗n β > ⃗n β +
2∑
i=1
< ∇⃗n α ,⃗e i > ⃗e i
150