Conformally Invariant Variational Problems. - SAM

written in isothermal coordinates. A further corollary of theorem
X.7 giving the Willmore surfaces equations in isothermal
coordinates is the following.
Corollary X.4. A conformal immersion Φ ⃗ of the flat disc D 2
is Willmore if and only if
[
div ∇H ⃗ −3π ⃗n (∇H)+⋆(∇ ⃗ ⊥ ⃗n∧ H) ⃗ ]
= 0 (X.86)
where the operators div, ∇ and ∇ ⊥ are taken with respect to the
flat metric 53 in D 2 .
✷
In order to exploit analytically equation (X.86) we will need
a more explicit expression of π ⃗n (∇ ⃗ H). Let be the interior
multiplication between p− and q−vectors p ≥ q producing p−
q−vectors in R m such that (see [Fe] 1.5.1 combined with 1.7.5)
: for every choice of p−, q− and p−q−vectors, respectively α,
β and γ the following holds
〈α β,γ〉 = 〈α,β ∧γ〉 .
Let (⃗e 1 ,⃗e 2 ) be an orthonormal basis of the orthogonal 2-plane
to the m−2 plane given by ⃗n and positively oriented in such a
way that
⋆(⃗e 1 ∧⃗e 2 ) = ⃗n
and let (⃗n 1···⃗n m−2 ) be a positively oriented orthonormal basis
of the m−2-plane given by ⃗n satisfying ⃗n = ∧ α ⃗n α . One verifies
easily that
⎧
⃗n ⃗e i = 0
⎪⎨
⎪⎩
⃗n ⃗n α = (−1) α−1 ∧ β≠α ⃗n β
⃗n (∧ β≠α ⃗n β ) = (−1) m+α−2 ⃗n α
53 divX = ∂ x1 X 1 +∂ x2 X 2 , ∇f = (∂ x1 f,∂ x2 f) and ∇ ⊥ f = (−∂ x2 f,∂ x1 f).
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