Conformally Invariant Variational Problems. - SAM

We have
∇ ⃗ Φ∧π T (∇ ⃗ H) = e λ [
< ⃗e 2 ,∂ x1
⃗ H > − < ⃗e1 ,∂ x2
⃗ H >
]
[
= e λ < π ⃗n (∂ x2 ⃗e 1 −∂ x1 ⃗e 2 ), H ⃗ ]
> ⃗e 1 ∧⃗e 2
= e
[< 2λ ⃗ I(⃗e 1 ,⃗e 2 )− ⃗ I(⃗e 2 ,⃗e 1 ), H ⃗ ]
> ⃗e 1 ∧⃗e 2
= 0
Therefore combining this identity with (X.193) gives
∇ ⃗ Φ∧⋆(∇ ⊥ ⃗n∧ ⃗ H) = 0
⃗e 1 ∧⃗e 2
(X.194)
(X.195)
We deduce from this equality that
∇ ⃗ Φ∧∇ ⊥ ⃗ L = ∇ ⃗ Φ∧∇ ⃗ H −3∇ ⃗ Φ∧π⃗n (∇ ⃗ H)
Combining this fact with (X.194) gives
∇ ⃗ Φ∧∇ ⊥ ⃗ L = −2∇ ⃗ Φ∧∇ ⃗ H
(X.196)
Itremainsnowtoexpress∇ ⃗ Φ∧∇ ⃗ H intermsofalinearcombinationofjacobiansinordertobe
ableto”factorize”thedivergence
operator in (X.196).
The definition of the contraction operation gives
Hence we have
⃗n ⃗ H =
m−2
∑
(−1) α−1 H α ∧ β≠α ⃗n β .
α=1
⋆(⃗n ⃗ H) =⃗e 1 ∧⃗e 2 ∧ ⃗ H .
(X.197)
We shall now compute ∇(⋆(⃗n ⃗ H)) ∇ ⊥ ⃗ Φ.
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