Conformally Invariant Variational Problems. - SAM

Denote e λ := |∂ x1Φ| ⃗ = |∂x2Φ|. ⃗ Then for any 0 < ρ < 1 there
exists a constant C ρ independent of Φ ⃗ such that
sup e λ (p) ≤ C ρ
[Area( ⃗ ] ( )
1/2
Φ(D 2 )) exp C |∇⃗n ⃗Φ |
p∈Bρ(0)
∫D 2 .
2 2
(X.165)
Moreover, for two given distinct points p 1 and p 2 in the interior
of D 2 and again for 0 < ρ < 1 there exists a constant C > 0
independent of Φ ⃗ such that
∫ ∣ ∣∣∣∣
‖λ‖ L∞ (Bρ 2(0)) ≤ C ρ |∇⃗n ⃗Φ | 2 +C ρ log |⃗ Φ(p 1 )− ⃗ Φ(p 2 )|
D |p 2 2 −p 1 | ∣
+C ρ log +[ C ρ Area( Φ(D ⃗ ]
2 )) .
where log + := max{log,0}.
Proof of theorem X.13. Denote
we have seen that
from which we deduce
( ⃗ f 1 , ⃗ f 2 ) = e −λ (∂ x1
⃗ Φ,∂x2 ⃗ Φ) .
−∇ ⊥ λ =< ⃗ f 1 ,∇ ⃗ f 2 > ,
(X.166)
✷
−∆λ =< ∂ x1
⃗ f1 ,∂ x2
⃗ f2 > − < ∂ x2
⃗ f1 ,∂ x1
⃗ f2 > .
(X.167)
Let (⃗e 1 ,⃗e 2 ) be the frame given by theorem X.8. There exists θ
such that such that e iθ (⃗e 1 +i⃗e 2 ) = ⃗ f 1 +i ⃗ f 2 and hence
and λ satisfies
< ∇ ⊥ ⃗ f1 ,∇ ⃗ f 2 >=< ∇ ⊥ ⃗e 1 ,∇⃗e 2 > .
−∆λ =< ∂ x1 ⃗e 1 ,∂ x2 ⃗e 2 > − < ∂ x2 ⃗e 1 ,∂ x1 ⃗e 2 > .
(X.168)
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