Conformally Invariant Variational Problems. - SAM

We then make the field ∇ ⊥ Bj i appear on the right-hand side by
observing that
(
∑n+1
n+1
) ∑
u j ∇u i·∇u j = ∇u i·∇ |u j | 2 /2 = ∇u i·∇|u| 2 /2 = 0 .
j=1
j=1
Deducting this null term from the right-hand side of (VII.24)
yields that for all i = 1···n+1, there holds
∑n+1
−∆u i = ∇ ⊥ Bj i ·∇uj
j=1
n+1
∑
= ∂ x Bj∂ i y u j −∂ y Bj∂ i x u i .
j=1
(VII.25)
We recognize the same Jacobian structure which we previously
employed to establish the regularity of solutions of the CMC
equation. It is thus possible to adapt mutatis mutandis our
argument to (VII.25) so as to infer that S n -valued harmonic
maps are regular.
VII.3 Hélein’s moving frames method and the regularity
of harmonic maps mapping into a manifold.
When the target manifold is no longer a sphere (or, more generally,
when it is no longer homogeneous), the aforementioned
Jacobian structure disappears, and the techniques we employed
no longer seem to be directly applicable.
Topalliatethislackofstructure,andthusextendtheregularityresulttoharmonicmapsmappingintoanarbitrarymanifold,
F. Hélein devised the moving frames method. The divergenceform
structure being the result of the global symmetry of the
target manifold, Hélein’s idea consists in expressing the harmonicmapequationinpreferredmovingframes,
calledCoulomb
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