Conformally Invariant Variational Problems. - SAM

Example 3. We consider a map (ω ij ) i,j∈Nm in C 1 (R m ,so(m)),
where so(m) is the space antisymmetricsquare m×m matrices.
We impose the following uniform bound
‖∇ω‖ L∞ (D 2 ) < +∞ .
For maps u ∈ W 1,2 (D 2 ,R m ), we introduce the Lagrangian
E ω (u) = 1 ∫ m∑
|∇u| 2 + ω ij (u)∂ x u i ∂ y u j −∂ y u i ∂ x u j dxdy
2 D 2 i,j=1
(VI.14)
The conformalinvariance of this Lagrangianarises from the fact
that E ω is made of the conformally invariant Lagrangian E to
whichisaddedtheintegraloverD 2 ofthe2-formω = ω i jdz i ∧dz j
pulled back by u. Composing u by an arbitrary positive diffeomorphism
of D 2 will not affect this integral, thereby making E ω
into a conformally invariant Lagrangian.
The Euler-Lagrange equation deriving from (VI.14) for variations
of the form u+tξ, where ξ is an arbitrary smooth function
with compact support in D 2 , is found to be
∆u i −2
m∑
Hkl(u) i ∇ ⊥ u k·∇u l = 0
k,l=1
∀ i = 1,...,m. (VI.15)
Here, ∇ ⊥ u l = (−∂ y u k ,∂ x u k ) 10 while Hkl i is antisymmetricin the
indices k et l. It is the coefficient of the R m -valued two-form H
on R m m∑
H i (z) := Hkl i (z) dzk ∧dz l .
k,l=1
10 in our notation, ∇ ⊥ u k ·∇u l is the Jacobian
∇ ⊥ u k ·∇u l = ∂ x u k ∂ y u l −∂ y u k ∂ x u l .
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