Conformally Invariant Variational Problems. - SAM

case it is a consequence of the following theorem due to Bang
Yen Chen [Che].
Theorem X.1. Let ⃗ Φ be the immersion of an n−dimensional
manifold Σ n into a riemannian manifold (M m ,g). Let µ be a
smooth function in M m and let h be the conformally equivalent
metric given by h := e 2µ g. We denote by ⃗ H g and ⃗ H h
the mean curvature vectors of the immersion ⃗ Φ respectively in
(M m ,g) and (M m ,h). We also denote by K g and K h the extrinsic
scalar curvatures respectively of (Σ n , ⃗ Φ ∗ g) and (Σ n , ⃗ Φ ∗ h).
With the previous notations the following identity holds
e 2µ ( | ⃗ H h | 2 h −Kh +K h) = | ⃗ H g | 2 g −Kg +K g . (X.21)
where K g (resp. K h ) is the sectional curvature of the subspace
⃗Φ ∗ T p Σ 2 in the manifold (M m ,g) (resp. (M m ,h)). K g − K g (
resp. K h −K h ) is also
✷
Remark X.1. K g − K g ( resp. K h − K h ) is also called the
extrinsic scalarcurvatureof the immersion ⃗ Φ os Σ n into (M m ,g)
(resp. into (M m ,h)).
Proof of theorem X.1.
Let ∇ g (resp. ∇ h ) be the Levi-Civita connection induced
by the metric g (resp. h) on M m . Let ∇ g (resp. ∇ h ) be the
Levi-Civita connection induced by the restriction of the metric
g (resp. h) on Σ n . By an abuse of notation we shall still write
g (resp. h) for the pull back by ⃗ Φ on Σ n of the restriction
of the metric g (resp. h). As in the previous section for any
vector X ∈ TΣ n we denote by ⃗ X the push forward by ⃗ Φ of X
: ⃗ X := d ⃗ Φ·X. With these notations, the second fundamental
form ⃗ I g of the immersion ⃗ Φ of Σ n into (M m ,g) at a point p ∈ Σ n
is defined as follows
∀X,Y ∈ T p Σ n
⃗ I g (X,Y) = ∇ g ⃗ Y
⃗ X −∇
g
Y X ,
114
(X.22)