Conformally Invariant Variational Problems. - SAM

Then ( ⃗ Φ,S, ⃗ R) satisfy the following system 75
⎧
⎪⎨
⎪⎩
∆S = − < ⋆∇⃗n,∇ ⊥ ⃗ R >
∆ ⃗ R = (−1) m ⋆(∇⃗n•∇ ⊥ ⃗ R)+∇ ⊥ S ⋆∇⃗n
∆ ⃗ Φ = 2 −1 ∇ ⊥ S ·∇ ⃗ Φ−2 −1 ∇ ⃗ R ∇ ⊥ ⃗ Φ
(X.223)
Remark X.4. The spectacular fact in (X.223) is that we have
deduced, from the Willmore equation, a system with quadratic
non-linearities which are made of linear combinations of jacobians.
We shall exploit intensively this fact below in order to
get the regularity of weak Willmore immersions and pass to the
limit in the system.
Proof of corollary X.5. The two first identitiesof (X.223)are
obtained by taking the divergence of (X.207).
Using the definition of the operation • and the second line of
(X.221), we have
∇ ⊥ Φ•∇ ⃗ R ⃗ =< ∇
⊥⃗ Φ, L ⃗ > ·∇Φ− ⃗ < ∇
⊥⃗ Φ,∇Φ ⃗ > L ⃗
+2∇ ⊥ Φ• ⃗
[(⋆(⃗n H)) ⃗ ∇Φ
⃗ ] (X.224)
It is clear that
< ∇ ⊥ ⃗ Φ,∇ ⃗ Φ >= 0 (X.225)
75 In codimension 1 the systems reads
⎧
⎪⎨
∆S = − < ∇⃗n,∇ ⊥ R ⃗ >
∆R ⃗ = ∇⃗n×∇ ⊥ R+∇ ⃗ ⊥ S∇⃗n
(X.222)
⎪⎩
∆ ⃗ Φ = ∇ ⊥ S∇ ⃗ Φ+∇ ⊥ ⃗ R×∇ ⃗ Φ
Observe that the use of two different operations, • in arbitrary codimension where ⃗ R is
seen as a 2-vector and × in 3 dimension ⃗ R is interpreted as a vector, generates formally
different signs. This might look first a bit confusing for the reader but we believed that
the codimension 1 case which more used in applications, deserved to be singled out.
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