Conformally Invariant Variational Problems. - SAM

the metric g. Here again, as in the 3-d case in the previous section,
from the fact that Levi-Civita connections are symmetric
we can deduce the symmetry of the second fundamental form.
Similarlyto the 3-d case, the mean curvature vector 38 is given
by
⃗H := 1 2 tr(g−1 ⃗
1
2∑
I) = g ij ⃗ I(∂xi ,∂ xj ) , (X.11)
2
ij=1
where (x 1 ,x 2 ) are arbitrary local coordinates in Σ 2 and (g ij ) ij is
the inverse matrix to (g(∂ xi ,∂ xj )).
We can now give the general formulation of the Willmore
energy of an immersion Φ ⃗ in R m of an abstract surface Σ 2 :
∫
W( Φ) ⃗ := | H| ⃗ 2 dvol g .
Σ 2
A fundamental theorem by Gauss gives an expression of the
intrinsic Gauss curvature at a point p ∈ Σ 2 in terms of the
second fundamental form of any immersion of the surface in
R m . Precisely this theorem says (see theorem 2.5 chapter 6 of
[doC2])
〈
K(p) = ⃗I(e1 ,e 1 ), ⃗ 〉
I(e 2 ,e 2 ) −〈
⃗I(e1 ,e 2 ), ⃗ 〉
I(e 1 ,e 2 ) (X.12)
where (e 1 ,e 2 ) is an arbitrary orthonormal basis of T p Σ 2 . From
this identity we deduce easily
| ⃗ I| 2 = 4| ⃗ H| 2 −2K . (X.13)
Hence, using Gauss bonnet theorem, we obtain the following
expression of the Willmore energy of an immersion into R m of
an arbitrary closed surface
W( Φ) ⃗ = 1 ∫
|
4
⃗ I| 2 dvol g +π χ(Σ 2 ) . (X.14)
Σ 2
38 observe that the notion of mean curvature H does not make sense any more in codimension
larger than 1 unless a normal direction is given.
110