Conformally Invariant Variational Problems. - SAM

The construction of conservation laws for systems
with antisymmetric potentials, and the proof of theorem
VIII.1.
The following result, combined to theorem VIII.3, implies
theorem VIII.1, itself yielding theorem VIII.2, and thereby providing
a proof of Hildebrandt’s conjecture, as we previously explained.
Theorem VIII.4. [Riv1] There exists a constant ε 0 (m) > 0
depending only on the integer m, such that for every vector field
Ω ∈ L 2 (D 2 ,so(m)) with
∫
|Ω| 2 < ε 0 (m) , (VIII.25)
D 2
it is possible to construct A ∈ L ∞ (D 2 ,Gl m (R))∩W 1,2 and B ∈
W 1,2 (D 2 ,M m (R)) with the properties
i)
∫
∫
|∇A| 2 +‖dist(A,SO(m))‖ L∞ (D 2 ) ≤ C(m) |Ω| 2 ,
D 2 D 2 (VIII.26)
ii)
div(∇ Ω A) := div(∇A−AΩ) = 0 ,
(VIII.27)
where C(m) is a constant depending only on the dimension m.
✷
Prior to delving into the proof of theorem VIII.4, a few comments
and observations are in order.
Glancingatthestatementofthetheorem,onequestionnaturally
arises: why is the antisymmetry of Ω so important?
It can be understood as follow.
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