Conformally Invariant Variational Problems. - SAM

where g(S) denotes the genus of S. Combining the definition of
W and this last identity one obtains 34
W(S)−π χ(S) = 1 ∫
(κ 2 1
4
+κ2 2 ) dvol g
S
= 1 ∫
|
4
⃗ I| 2 dvol g = 1 ∫
|d⃗n| 2 g dvol g .
S 4 S
(X.7)
Hence modulo the addition of a topological term, the Willmore
energy corresponds to the Sobolev homogeneous Ḣ1 −energy of
the Gauss map for the induced metric g.
Consider now, again for a closed surface, the following identity
based again on Gauss-Bonnet theorem (X.6) :
W(S) = 1 ∫ ∫
(κ 1 −κ 2 ) 2 dvol g + κ 1 κ 2 dvol g
4 S
S
= 1 ∫
(X.8)
(κ 1 −κ 2 ) 2 dvol g +2π χ(S) .
4
S
Hence, modulo this time the addition of the topological invariant
of the surface 2πχ(S), the Willmore energy identifies with
an energy that penalizes the lack of umbilicity when the two
principal curvatures differ. The Willmore energy in this form is
commonly called Umbilic Energy.
Finally there is an interesting last expression of the Willmore
energy that we would like to give here. Consider a conformal
parametrization ⃗ Φ of the surface S for the conformal structure
induced by the metric g (it means that we take a Riemann surface
Σ 2 and a conformal diffeomorphism ⃗ Φ from Σ 2 into S). In
34 The last identity comes from the fact that, at a point p, taking an orthonormal basis
(⃗e 1 ,⃗e 2 ) of T p S one has, since < d⃗n,⃗n >= 0 :
|d⃗n| 2 g = 2∑
i,j=1
< d⃗n· ⃗e i ,⃗e j > 2 =
2∑
| ⃗ I(⃗e i ,⃗e j )| 2 = | ⃗ I| 2 .
i,j=1
105