Conformally Invariant Variational Problems. - SAM

Proof of theorem X.2 Denote by NΣ 2 the pullback by Φ ⃗
of the normal sphere bundle to Φ(Σ ⃗ 2 ) in R m made of the unit
normal vectors to Φ ⃗ ∗ TΣ 2 . Denote by G ⃗ = (G 1 ,··· ,G m ) the
map from NΣ 2 into S m−1 the unit sphere which to an element
in NΣ 2 assigns the corresponding unit vector in S m−1 . For any
k ∈ N, k ≥ 1, we denote
Observe that
ω S
k−1 = 1
|S k−1 |
∫
k∑
(−1) j−1 x j ∧ l≠j dx l .
j=1
S k−1 ω S
k−1 = 1 .
Locally on Σ 2 we can choose an orthonormal positively oriented
frame of the normal plane : (⃗n 1···⃗n m−2 ). We also chose locally
an orthonormal tangent frame (⃗e 1 ,⃗e 2 ). This normal frame permits
locally, over an open disk U ⊂ Σ 2 , to trivialize NΣ 2 and
the map ⃗ G can be seen as a map from U ×S m−3 into S m−1 :
⃗G(p,s) =
m−2
∑
α=1
s α ⃗n α (p) ,
where ∑ α s2 α = 1. Let p ∈ U, in order to simplify the notations
we may assume that (⃗e 1 ,⃗e 2 ,⃗n 1···⃗n m−2 ) at p coincides with the
canonical basis of R m . Hence we have G 1 (p) = G 2 (p) = 0 and
G α+2 (p) = s α for α = 1···m−2.
and
dG 1 (p) =
dG 2 (p) =
m−2
∑
α=1
m−2
∑
α=1
∀ α = 1···m−2 dG α+2 = ds α +
s α < d⃗n α ,⃗e 1 > ,
s α < d⃗n α ,⃗e 2 > ,
122
m−2
∑
β=1
s β < d⃗n β ,⃗n α > .