Conformally Invariant Variational Problems. - SAM

local conformal coordinates (x 1 ,x 2 ) one has ∂ x1
⃗ Φ · ∂x2
⃗ Φ = 0,
|∂ x1
⃗ Φ| 2 = |∂ x2
⃗ Φ| 2 = e 2λ and the mean-curvature vector is given
by
⃗H = e−2λ
2 ∆⃗ Φ = 1 2 ∆ ⃗ gΦ ,
where∆denotesthenegative”flat”laplacian,∆ = ∂ 2 +∂ 2 and
x 2 1 x2, 2
∆ g is the intrinsic negative Laplace-Beltrami operator. With
these notations the Willmore energy becomes
W(Σ 2 ) = 1 ∫
|∆ gΦ| ⃗ 2 dvol g . (X.9)
4 Σ 2
HencetheWillmoreenergyidentifiesto1/4−thoftheBi-harmonic
Energy 35 of any conformal parametrization ⃗ Φ.
X.2 The role of Willmore energy in different areas of
sciences and technology.
As mentioned in the beginning of the present chapter, Willmore
energy has been first considered in the framework of the modelization
of elastic surfaces and was proposed in particular by
Poisson [Poi] in 1816 as a Lagrangian from which the equilibrium
states of such elastic surfaces could be derived. Because
of it’s simplicity and the fundamental properties it satisfies the
Willmore energy has appeared in many area of science for two
centuries already. One can quote the following fields in which
the Willmore energy plays an important role.
- Conformal geometry : Because of it’s conformal invariance
(see the next section), the Willmore energy has been introduced
in the earlyXX-thcentury-see the book of Blaschke
[Bla3] - as a fundamental tool in Conformal or Möbius geometry
of submanifolds.
35 This formulation has also the advantage to show clearly why Willmore energy is a
4-th order elliptic problem of the parametrization.
106