Conformally Invariant Variational Problems. - SAM

There exists a constant C > 0, such that, if one assumes that
∫
|∇⃗n| 2 < 8π , (X.146)
D 3
2
then there exists ⃗e 1 and ⃗e 2 in W 1,2 (D 2 ,S m−1 ) such that
⃗n = ⋆(⃗e 1 ∧⃗e 2 ) ,
(X.147)
and 60
∫
D 2
2∑
|∇⃗e i | 2 dx 1 dx 2 ≤ C
i=1
∫
D 2 |∇⃗n| 2 dx 1 ,dx 2 .
(X.148)
✷
The requirement for the L 2 nom of ∇⃗n to be below a threshold,
inequality (X.146), is necessary and it is conjectured in [He]
that 8π/3 could be replaced by 8π and that this should be optimal.
We can illustrate the need to have an energy restriction such
as (X.146) with the following example :
Let π be the stereographic projection from S 2 into C∪{∞}
which sends the north pole N = (0,0,1) to 0 and the south pole
S = (0,0,−1) to ∞. For λ sufficiently small we consider on D 2
the following map ⃗n λ taking value into S 2 :
⎧
⃗n ⎪⎨ λ (x) := π −1 (λx) for |x| ≤ 1/2
⎪⎩
⃗n λ (x) := (1−r) π−1 (λx)+(r−1/2) S
|(1−r) π −1 (λx)+(r−1/2) S|
for 1 2
< |x| ≤ 1.
The map ⃗n λ has been constructed in such a way that on B 1/2 (0)
itcoversthemostpartofS 2 conformally 61 andthemissingsmall
60 The condition (X.147) together with the fact that the ⃗e i are in S m−1 valued imply,
since ⃗n has norm one, that ⃗e 1 and ⃗e 2 are orthogonal to each other.
61 The map x → π −1 (λx) is conformal.
165