Conformally Invariant Variational Problems. - SAM

The definition of the Gauss curvature gives
K ⃗n = − e−2λ
2
∇⃗n×∇ ⊥ ⃗n = − e−2λ
2
Inserting (X.95) in (X.94) gives
−4e 2λ ⃗ H (H 2 −K)+∇H ·[∇⃗n+⃗n×∇ ⊥ ⃗n ]
This becomes
= Hdiv [ ∇⃗n−⃗n×∇ ⊥ ⃗n ] .
−4e 2λ H ⃗ (H 2 −K)−2∆H ⃗n
[
= div −2∇H ⃗n+H ∇⃗n− H ⃗ ]
×∇ ⊥ ⃗n
div [ ⃗n×∇ ⊥ ⃗n ] (X.95)
.
(X.96)
(X.97)
Usingnow”intrinsicnotations”(independentoftheparametrization)
on (Σ 2 ,g), (X.97) says
4 H ⃗ (H 2 −K)+2∆ g H ⃗n
[
= d ∗ g
−2dH ⃗n+Hd⃗n−∗ H ⃗ ]
×d⃗n
.
(X.98)
which is the desired identity (X.85) and theorem X.7 is proved
in codimension 1.
✷
Before to proceed to the proof of theorem X.7 in arbitrary
codimensionwe first introducesome complexnotationsthatwill
be useful in the sequel. Assume Φ ⃗ is a conformal immersion
into R m , one denotes z = x 1 + ix 2 , ∂ z = 2 −1 (∂ x1 − i∂ x2 ), ∂ z =
2 −1 (∂ x1 +i∂ x2 ).
Moreover we denote 55
⎧
⎨ ⃗e z := e −λ ∂ zΦ ⃗ = 2 −1 (⃗e 1 −i⃗e 2 )
⎩
⃗e z := e −λ ∂ zΦ ⃗ = 2 −1 (⃗e 1 +i⃗e 2 )
55 Observe that the notation has been chosen in such a way that ⃗e z =⃗e z .
148