Conformally Invariant Variational Problems. - SAM

the curl operator and finally subtracting some harmonic R (for
S) or ∧ 2 R (for ⃗ R) valued map.
It remains to prove (X.207). Let ⃗ N be a normal vector, exactly
like for the particular case of ⃗ H in (X.197)
(−1) m−1 ⋆(⃗n ⃗ N) =⃗e 1 ∧⃗e 2 ∧ ⃗ N .
(X.210)
We deduce from this identity
(−1) m−1 (⋆(⃗n ⃗ N)) ∇ ⃗ Φ = (⃗e 1 ∧⃗e 2 ∧ ⃗ N) ∇ ⃗ Φ = −∇ ⊥ ⃗ Φ∧ ⃗ N .
(X.211)
Applying this identity to ⃗ N := ⃗ H implies
∇ ⃗ R = ⃗ L∧∇ ⃗ Φ−2(−1) m−1 ∇ ⊥ ⃗ Φ∧ ⃗ H
(X.212)
We takenowthe•contractionbetween⃗nand∇ ⃗ R andweobtain
⃗n•∇ ⃗ R = (⃗n ⃗ L)∧∇ ⃗ Φ+2(−1) m−1 (⃗n ⃗ H)∧∇ ⊥ ⃗ Φ
= (⃗n π ⃗n ( ⃗ L))∧∇ ⃗ Φ+2(−1) m−1 (⃗n ⃗ H)∧∇ ⊥ ⃗ Φ .
(X.213)
For a normal vector ⃗ N again a short computation gives
⋆[(⃗n ⃗ N)∧∇ ⃗ Φ] = (−1) m ∇ ⊥ ⃗ Φ∧ ⃗ N ,
(X.214)
from which we also deduce
⋆[(⃗n ⃗ N)∧∇ ⊥ ⃗ Φ] = (−1) m−1 ∇ ⃗ Φ∧ ⃗ N .
(X.215)
Combining (X.213), (X.214) and (X.215) gives then
⋆(⃗n•∇ ⃗ R) = −∇ ⊥ ⃗ Φ∧π⃗n ( ⃗ L)+2∇ ⃗ Φ∧ ⃗ H .
(X.216)
from which we deduce
⋆(⃗n•∇ ⊥ ⃗ R) = −π⃗n ( ⃗ L)∧∇ ⃗ Φ+2∇ ⊥ ⃗ Φ∧ ⃗ H .
(X.217)
Combining (X.211) and (X.217) gives
(−1) m ⋆(⃗n•∇ ⊥ ⃗ R) = ∇ ⃗ R+(−1) m π T ( ⃗ L)∧∇ ⃗ Φ . (X.218)
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