Conformally Invariant Variational Problems. - SAM

iemannian manifold (M m ,g). Let Ψ be a positive conformal
diffeomorphism from (M m ,g) into another riemannian oriented
manifold (N m ,k) then we have the following pointwise identity
everywhere on Σ 2
[
| H ⃗ Φ ⃗ ∗g | 2 Φ
−K ⃗ ∗g +K g] dvol ⃗Φ∗ g
[
= | H ⃗ (Ψ◦⃗ Φ) ∗k | 2 −K (Ψ◦⃗ Φ) ∗k +K k] dvol (Ψ◦Φ)∗ ⃗ k
(X.47)
where K g (resp. K k ) is the sectional curvature of the 2−plane
⃗Φ ∗ TΣ 2 in (M m ,g) (resp. of the two plane Ψ ∗Φ∗ ⃗ TΣ 2 in (N m ,k)).
In particular the following equality holds
∫
W( Φ)+ ⃗ K g dvol ⃗Φ∗ g
Σ 2 ∫
(X.48)
= W(Ψ◦Φ)+
⃗ K k dvol (Ψ◦Φ)∗ ⃗ g
.
Σ 2
Proof of corollary X.1.
By definition Ψ realizes an isometry between (M m ,Ψ ∗ k) and
(N m ,k). Let µ ∈ R such that e µ g = Ψ ∗ k. We can apply the
previous theorem with h = Ψ ∗ k and we obtain
[
| ⃗ H ⃗ Φ ∗g | 2 −K ⃗ Φ ∗ g
]
= e 2µ [
| ⃗ H (Ψ◦⃗ Φ) ∗k | 2 −K (Ψ◦⃗ Φ) ∗ k
]
✷
. (X.49)
Itisalsoclearthatthevolumeformdvol g givenbytherestriction
to Σ 2 of the metric g is equal to e −2µ dvol Ψ∗ k where dvol Ψ∗ k is
equal to the volume form given by the restriction to Σ 2 of the
metric Ψ ∗ k. Hence combining this last fact with (X.47). (X.48)
is obtained by integrating (X.47) over Σ 2 , the scalar curvature
terms canceling each other on both sides of the identity due to
Gauss Bonnet theorem. Corollary X.1 is then proved. ✷
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