Conformally Invariant Variational Problems. - SAM

ii)
iii)
iv)
limsupArea( Φ ⃗ k (D 2 )) < +∞
k→+∞
∀ρ < 1 limsup‖log|∇Φ ⃗ k |‖ L∞ (Bρ 2
k→+∞
(0)) < +∞ ,
div
[
∇H ⃗ k −3π ⃗nk (∇H ⃗ k )+⋆(∇ ⊥ ⃗n k ∧H ⃗ ]
k ) = 0 .
Moreover we can assume that the immersion does not shift to
infinity by taking
⃗Φ k (0) = 0 . (X.178)
Since the immersions Φ ⃗ k are conformal we have that
∆Φ ⃗ k = 2e 2λ k Hk ⃗
where e λ k = |∂ ⃗
x 1Φk | = |∂ x2Φk ⃗ |.
The conditions i) and iii) imply then the following
∀ρ < 1 limsup‖∆Φ ⃗ k ‖ L2 (Bρ(0)) < +∞ .
2
k→+∞
Moreover condition ii) implies
∫
limsup |∇Φ ⃗ k | 2 = limsup
k→+∞ D 2 k→+∞
2
∫
D 2 e 2λ k
= limsup2Area( Φ ⃗ k (D 2 )) < +∞ .
k→+∞
Combining (X.178), (X.179) and (X.180) we obtain
∀ρ < 1 limsup‖ Φ ⃗ k ‖ W
2,2
(Bρ 2
k→+∞
(0)) < +∞
(X.179)
(X.180)
(X.181)
We can then extract a subsequence that we keep denoting ⃗ Φ k
such that there exists a map ⃗ Φ ∞ ∈ W 2,2
loc (D2 ,R m ) for which
⃗Φ k ⇀ ⃗ Φ ∞ weakly in W 2,2
loc (D2 ,R m ) .
181