Conformally Invariant Variational Problems. - SAM

makes nonlinearities of the form
A(u)(∇u,∇u)−H(u)(∇ ⊥ u,∇u) ,
sospecialandmorefavorabletotreatingregularity/compactness
matters than seemingly simpler nonlinearities, such as
|∇u| 2 ,
which we encountered in Section 1?
The moving frames method does not address this question.
We consider a weakly harmonic map u with finite energy, on
D 2 and taking values in a regular oriented closed submanifold
N n ⊂ R n+1 of codimension 1. We saw at the end of Section 2
that u satisfies the equation
−∆u = ν(u) ∇(ν(u))·∇u ,
(VIII.2)
where ν is the normal unit-vector to N n relative to the orientation
of N n .
In local coordinates, (VIII.2) may be recast as
∑n+1
−∆u i = ν(u) i ∇(ν(u)) j ·∇u j ∀ i = 1···n+1 .
j=1
(VIII.3)
Inthismoregeneralframework,wemayattempttoadaptHélein’s
operation which changes (VII.24) into (VII.25). The first step
of this process is easily accomplished. Indeed, since ∇u is orthogonal
to ν(u), there holds
∑n+1
ν j (u)∇u j = 0 .
j=1
Substituting this identity into (VIII.4) yields another equivalentformulationofthe
equationsatisfiesby N n -valuedharmonic
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