Conformally Invariant Variational Problems. - SAM

Combining (X.14), (X.16) and (X.18) we obtain
W( Φ) ⃗ = 1 ∫
|d⃗n| 2 g dvol g +π χ(Σ 2 ) ,
4 Σ 2
(X.19)
whichgeneralizestoarbitraryimmersionsofclosed2-dimensional
surfaces the identity (X.7).
X.3.2 The WillmoreEnergy ofimmersions into ariemannian Manifold
(M m ,h).
Let Σ n be an abstract n−dimensional oriented manifold. Let
(M m ,g)beanarbitraryriemannianmanifoldofdimensionlarger
or equal to n + 1. The Willmore energy of an immersion ⃗ Φ
into (M m ,g) can be defined in a similar way as in the previous
subsection by formally replacing the exterior differential d with
the Levi-Civita connection ∇ on M induced by the ambient
metric g. Precisely we denote still by g the pull-back of the
ambient metric g by ⃗ Φ.
∀p ∈ Σ 2 ∀X,Y ∈ T p Σ 2
g(X,Y) := g(d ⃗ Φ·X,d ⃗ Φ·Y)
The volume form associated to g on Σ 2 at the point p is still
given by
√
dvol g := det(g(∂ xi ,∂ xj )) dx 1 ∧···∧dx n ,
where (x 1 ,··· ,x n ) are arbitrary local positive coordinates. The
second fundamental form associated to the immersion ⃗ Φ at a
point p of Σ n is the following map
⃗ Ip : T p Σ n ×T p Σ n −→ ( ⃗ Φ ∗ T p Σ n ) ⊥
(X,Y)
−→ ⃗ I p (X,Y) := π ⃗n (∇ ⃗Y (d ⃗ Φ·X))
where π ⃗n denotes the orthogonal projection from T ⃗Φ(p) M onto
the space orthogonal to ⃗ Φ ∗ (T p Σ n ) with respect to the g metric
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