Conformally Invariant Variational Problems. - SAM

We make now use of (X.87) and we deduce
π ⃗n (∇ ⊥ ⃗ fi ) = (−1) m−1 ⃗n (⃗n ∇ ⊥ ⃗ fi )
= ∇ ⊥ (π ⃗n ( ⃗ f i ))+(−1) m−1 ∇ ⊥ ⃗n (⃗n ⃗ f i )
(X.155)
+(−1) m−1 ⃗n (∇ ⊥ ⃗n ⃗ f i )
Using the fact that π ⃗n ( f ⃗ i ) ≡ 0 we obtain from (X.155) that
∫
∫
|π ⃗n (∇f ⃗ i )| 2 dx 1 dx 2 ≤ 2 |∇⃗n| 2 dx 1 dx 2 (X.156)
D 2 D 2
Combining (X.154) and (X.156) we obtain
∫
∫
| < ∇ ⊥ f1 ⃗,∇f ⃗ 2 > | dx 1 dx 2 ≤ 2 |∇⃗n| 2 dx 1 dx 2 (X.157)
D 2 D 2
This estimate together with standard elliptic estimates (see for
instance [Ad]) give
‖ < f ⃗ 1 ,∇f ⃗ 2 > ‖ L
2,∞
(D 2 ) = ‖∇λ‖ L
2,∞
(D 2 )
∫
≤ C |∇⃗n| 2 dx 1 dx 2 .
D 2
(X.158)
From (X.156) and (X.158) we deduce the following theorem 64
.
Theorem X.10. Let ⃗n in W 1,2 (D 2 ,Gr m−2 (R m )), then there exists
⃗e 1 ,⃗e 2 ∈ W 1,2 (D 2 ,S m−1 ) such that
⃗e 1 ∧⃗e 2 = ⋆⃗n ,
div = 0 ,
64 Similarly it would be interesting to explore the possibility to construct global gauges
with estimates in non abeliangaugetheory. Forinstance onecanaskthe followingquestion
: for a given curvature of a W 1,2 SU(n)−connection over the 4-dimensional ball, can one
construct a gauge in which the L 4,∞ norm of the connection is controlled by the L 2 −norm
of the curvature ?
169