Conformally Invariant Variational Problems. - SAM

present in the original nomenclature given by Wilhelm Blaschke
and his school where Willmore surfaces were called conformal
minimal surfaces see [Bla3] and [Tho]. Because of this strong
link with minimal surface theory combined with the fundamental
conformal invariance property one can then naturally expect
the family of Willmore surfaces to be of special interest in geometry
.
X.5.1 The Euler-Lagrange equation of Shadow, Thomsen and Weiner.
We first introduce the notion of Willmore surfaces.
Definition X.3. Let ⃗ Φ be a smooth immersion of a surface Σ 2
such that W( ⃗ Φ) < +∞. ⃗ Φ is a critical point for W if
∀ ⃗ ξ ∈ C ∞ 0 (Σ 2 ,R m )
Such an immersion is called Willmore.
d
dt W(⃗ Φ+t ⃗ ξ) t=0 = 0
(X.63)
Willmoreimmersionsare characterizedby an Euler Lagrange
equation which has been discovered in dimension 3 by Shadow 46
and also appear in the PhD work of Gerhard Thomsen [Tho],
student of Wilhelm Blaschke. The equation in general codimension,
m ≥ 3 arbitrary has been derived by Joel Weiner in [Wei].
Theorem X.5. A smooth immersion Φ ⃗ into R m is Willmore
(i.e. satisfies (X.63)) if and only if it solves the following equation
∆ ⊥H⃗Φ ⃗ −2| H ⃗ ⃗Φ | 2 H⃗Φ ⃗ +Ã(⃗ H ⃗Φ ) = 0 (X.64)
where ∆ ⊥ is the negative covariant laplacian operator for the
connection 47 induced by the ambiant metric on the normal bun-
46 See the comment by Whilhelm Blaschke in Ex. 7 83 chapter 8 of [Bla3].
47 The assocated covariant derivative for any normal vector-field ⃗ X is given by
D Z
⃗ X := π⃗n (d ⃗ X ·Z) .
✷
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