Conformally Invariant Variational Problems. - SAM

orthonormal basis for the metric g. Hence the corresponding
expression to (X.33) for the mean curvature vector ⃗ H h of the
same immersion ⃗ Φ but into (M m ,h) reads
⃗H h = e−µ
n
m−2
∑
α=1
n∑
κ h i(e −µ ⃗n α ) ⃗n α .
i=1
(X.42)
A vector e ∈ T p Σ 2 \{0} is an eigenvector for g(⃗n α , ⃗ I g (·,·)) w.r.t.
the metric g if and only if there exists a real number κ such that
g(⃗n α , ⃗ I g (·,e)) = κ g(·,e) .
(X.43)
This implies that
h(⃗n α , ⃗ I h (·,e)+g(·,e) ⃗ W) = κ h(·,e) .
In other words we have obtained
h(e −µ ⃗n α , ⃗ [ I h (·,e)) = e −µ κ−g(⃗n α , W) ⃗ ]
h(·,e) .
(X.44)
This later identity says then that e is also an eigenvector of
h(e −µ ⃗n α , ⃗ I h (·,·)) with eigenvalue e
[κ−g(⃗n −µ α , W) ⃗ ]
. We then
have
[ κ h i (e−µ ⃗n α ) = e −µ κ g i (⃗n α)−g(⃗n α , W) ⃗ ]
. (X.45)
This implies that ∀α ∈ {1···m−2} and ∀i,j ∈ {1···n}
∣ κ
g
i (⃗n α)−κ g j (⃗n α) ∣ 2 = e ∣ 2µ κ
h
i (e −µ ⃗n α )−κ h j(e −µ ⃗n α ) ∣ 2
. (X.46)
Combining (X.41) (which is also valid for h of course) with
(X.46) gives (X.21) and theorem X.1 is proved. ✷
We will make use of the following corollary of theorem X.1.
Corollary X.1. Let Σ 2 be a closedsmoothoriented2-dimensional
manifold and let ⃗ Φ be an immersion of Σ 2 into an oriented
119