Conformally Invariant Variational Problems. - SAM

and the following equation holds
⎧
⎨ ∇S = − < ⋆⃗n,∇ ⊥ R ⃗ >
⎩
∇R ⃗ = (−1) m ⋆(⃗n•∇ ⊥ R)+(−1) ⃗ m−1 ∇ ⊥ S ⋆⃗n ,
(X.207)
where • is the following contraction operation which to a pair of
respectively p− and q−vectors of R m assigns a p+q−2−vector
of R m such that
∀⃗a ∈ ∧ p R m
∀ ⃗ b ∈ ∧ 1 R m ⃗a• ⃗ b :=⃗a ⃗ b
and
∀⃗a ∈ ∧ p R m ∀ ⃗ b ∈ ∧ r R m ∀⃗c ∈ ∧ s R m
⃗a•( ⃗ b∧⃗c) := (⃗a• ⃗ b)∧⃗c+(−1) rs (⃗a•⃗c)∧ ⃗ b
✷
Remark X.3. In the particular case m = 3 both ⃗n and R ⃗ can be
identifiedwithvectors by the meanof the Hodgeoperator⋆. Once
this identification is made the systems (X.206) and (X.207) become
respectively
⎧
⎨ ∇S =< L,∇ ⃗ Φ ⃗ >
⎩
∇R ⃗ = L×∇ ⃗ Φ+2 ⃗ H ∇Φ ⃗ (X.208)
.
and
⎧
⎨
⎩
∇S = −
∇ ⃗ R = ⃗n×∇ ⊥ ⃗ R+∇ ⊥ S ⃗n ,
(X.209)
Proof of theorem X.15. The existence of S and ⃗ R satisfying
(X.206) is obtained exactly like for ⃗ L in the beginning of
the proof of theorem X.14 taking successively the convolution of
the divergence free quantities with −(2π) −1 log r, then taking
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