Conformally Invariant Variational Problems. - SAM

a nonlinear Hodge decomposition 25 in L 2 of Ω. Thus, let
ξ ∈ W 1,2 (D 2 ,so(m)) and P be a W 1,2 map taking values in
the group SO(m) of proper rotations of R m , such that
Ω = P ∇ ⊥ ξP −1 −∇P P −1 .
(VIII.31)
At first glance, the advantage of (VIII.31) over (VIII.30) is not
obvious. If anything, it seems as though we have complicated
the problem by having to introduce left and right multiplications
by P and P −1 . On second thought, however, since rotations
are always bounded, the map P in (VIII.31) is an element
of W 1,2 ∩ L ∞ , whereas in (VIII.30), the map P belonged only
to W 1,2 . This slight improvement will actually be sufficient to
successfully carry out our proof. Furthermore, (VIII.31) has yet
another advantage over (VIII.30). Indeed, whenever A and B
are solutions of (VIII.27), there holds
∇ ∇⊥ ξ(AP) = ∇(AP)−(AP)∇ ⊥ ξ
Hence, via setting
= ∇AP +A∇P −AP (P −1 ΩP +P −1 ∇P)
= (∇ Ω A)P = −∇ ⊥ B P .
à := AP, Ã, we find
∆à = ∇÷∇⊥ ξ +∇ ⊥ B ·∇P .
(VIII.32)
Unlike (VIII.30), the second summand on the right-hand side of
(VIII.32) is a linear combination of Jacobians of terms which lie
inW 1,2 . Accordingly,callingupontheoremVII.1,wecancontrol
à in L ∞ ∩ W 1,2 . This will make a bootstrapping argument
possible.
One point still remains to be verified. Namely, that the nonlinear
Hodge decomposition (VIII.31) does exist. This can be
accomplished with the help of a result of Karen Uhlenbeck 26 .
25 which is tantamount to a change of gauge.
26 In reality, this result, as it is stated here, does not appear in the original work of Uhlenbeck.
In [Riv1], it is shown how to deduce theorem VIII.5 from Uhlenbeck’s approach.
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