Conformally Invariant Variational Problems. - SAM

dle to Φ(Σ ⃗ 2 ) : for all X ⃗ normal vector field to Φ(Σ ⃗ 2 ) one has
[
∆ ⊥X ⃗ := −π⃗n
[d ∗ g
(π ⃗n dX
⃗ ])]
[ (
= π ⃗n (detg) −1/2 ∂ xi g ij√ [ ])]
detg π ⃗n ∂ xjX ⃗
where d ∗ g
is the adjoint of d for the induced scalar product g :=
⃗Φ ∗ g R
m on Σ 2 and where we are using local coordinates on Σ 2 in
the last line. Finally à is the following linear map
∀ ⃗ X ∈ R m Ã( ⃗ X) =
2∑
⃗ I(ei ,e j ) < ⃗ I(e i ,e j ), X ⃗ > , (X.65)
i,j=1
where (e 1 ,e 2 ) is an orthonormal basis of TΣ 2 for the induced
metric g := Φ ⃗ ∗ g R m.
✷
In the sequel we shall often omit the suscribt ⃗ Φ when there
is no ambiguity and for instance ⃗ H ⃗Φ will sometimes be simply
denoted ⃗ H.
In dimension 3 the equation (X.64) takes a simpler form. ⃗n
in this case is an S 2 valued vector and
π ⃗n (dH) ⃗ 〈
= ⃗n,d H ⃗ 〉
⃗n = dH ⃗n .
where ∗ g is the Hodge operator for the induced metric g on Σ 2
pulled-back of the ambiant metric in R m by Φ. ⃗ Hence
(
d ∗ g
π ⃗n (dH)
⃗ )
= d ∗ g
(dH ⃗n) = d ∗ g
dH ⃗n+∗ g (dH ∧∗ g d⃗n) .
Thus
∆ ⊥
⃗ H = −π⃗n
[d ∗ g
(
π ⃗n (dH)
⃗ )]
= −d ∗ g
dH ⃗n = ∆ g H ⃗n , (X.66)
where ∆ g is the negative Laplace Beltrami operator for the induced
metric g on Σ 2 . Since ⃗ I takes value in the normal plane
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