Conformally Invariant Variational Problems. - SAM

Observe that π ⃗n (∂ x1 ⃗e 1 ) + π ⃗n (∂ x2 ⃗e 2 ) = 2 e λ ⃗ H hence finally we
obtain
∇(⋆(⃗n ⃗ H)) ∇ ⊥ ⃗ Φ = ∇ ⃗ Φ∧∇ ⃗ H .
(X.203)
Combining (X.196) and (X.203) gives
∇ ⃗ Φ∧∇ ⊥ ⃗ L = −2∇(⋆(⃗n ⃗ H)) ∇
⊥⃗ Φ
(X.204)
This is exactly the conservation law (X.188) and theorem X.14
is proved.
✷
Having now found 2 new conserved quantities, as we did for
thefirstone∇ ⃗ H−3π ⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H),wecanapplyPoincaré
lemmain ordertoobtain”primitives”of these quantities. These
”primitive” quantities will satisfy a very particular elliptic system.
Precisely we have the following theorem.
Theorem X.15. Let Φ ⃗ be a conformal lipschtiz immersion of
the disc D 2 with L 2 −bounded second fundamental form. Assume
there exists L ⃗ ∈ L 2,∞ (D 2 ,R m ) satisfying
⎧
⎪⎨ div < L,∇ ⃗ ⊥ Φ ⃗ >= 0
[ ] (X.205)
⎪⎩ 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. ⃗ There exists 74 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.206)
.
74 We denote by W 1,(2,∞) the space of distribution in L 2 with gradient in L 2,∞ .
190