Conformally Invariant Variational Problems. - SAM

One verifies easily that
π T ( ⃗ L)∧∇ ⃗ Φ = ∇ ⊥ S ⃗e 1 ∧⃗e 2
= ∇ ⊥ S ⋆⃗n
(X.219)
The combination of (X.218) and (X.219) gives the second equation
of(X.207). The first equation is obtained by taking the
scalar product between the first equation and ⋆⃗n once one has
observed that
< ⋆⃗n,⋆(⃗n•∇ ⊥ ⃗ R) >= 0
Thislaterfactcomesfrom(X.216)whichimpliesthat⋆(⃗n•∇ ⊥ R) ⃗
is a linear combination of wedges of tangent and normal vectors
to Φ ⃗ ∗ TD 2 . Hence theorem X.15 is proved. ✷
An important corollary of the previous theorem is the following.
Corollary X.5. Let Φ ⃗ be a conformal lipschitz immersion of
the disc D 2 with L 2 −bounded second fundamental form. Assume
there exists L ⃗ in L 2,∞ (D 2 ,R m ) satisfying
⎧
⎪⎨ div < L,∇ ⃗ ⊥ Φ ⃗ >= 0
[ ] (X.220)
⎪⎩ div ⃗L∧∇ ⊥⃗ Φ+2 (⋆(⃗n H)) ⃗ ∇
⊥⃗ Φ = 0 .
where H ⃗ and ⃗n denote respectively the mean-curvature vector
and the Gauss map of the immersion Φ. ⃗ Let S ∈ W 1,(2,∞)
loc
(D 2 ,R)
and R ⃗ ∈ W 1,(2,∞)
loc
(D 2 ,∧ 2 R m ) such that
⎧
⎨ ∇S =< L,∇ ⃗ Φ ⃗ >
⎩
∇R ⃗ = L∧∇ ⃗ Φ+2 ⃗ (⋆(⃗n H)) ⃗ ∇Φ ⃗ (X.221)
.
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