Conformally Invariant Variational Problems. - SAM

is an immersion or not. A first observation in this direction can
be made. The harmonicity of u (V.4) says that div(∇u k ) = 0
on D 2 for each component u k of u. Applying Poincaré lemma,
we can introduce the harmonic conjugates v k of u k satisfying
(−∂ x2 v k ,∂ x1 v k ) = ∇ ⊥ v k := ∇u k = (∂ x1 u k ,∂ x2 u k )
This implies that f k (z) := u k −iv k is holomorphic and
|f ′ (z)| 2 = 4 −1 [|∂ x1 u k −∂ x2 v k | 2 +|∂ x2 u k +∂ x1 v k | 2 ] = |∇u k | 2 .
Since u is conformal, it is an immersion at a given point (i.e
|du ∧ du| ≠ 0) if and only if |∇u| ≠ 0 which is then equivalent
to |f ′ (z)| ≠ 0. Since f ′ (z) is also holomorphic, we obtain
that u is a-priori is an immersion away from isolated so called
branched points. In other words u is is what is called a branched
immersion.
Finally one has to study the possibility for the branch points
to exist or not. It is clear that in codimension larger than 1,
such a branch point can exist as one can see by taking a sub
part of a complex algebraic curve in R 4 ≃ C 2 such as
z → (z 2 ,z 3 )
This disc is calibrated by the standard Kähler form of C 2 and is
then area minimizing for its boundary data, the curve Γ given
by e iθ → (e 2iθ ,e 3iθ ).
In codimension1, m = 3, however the situationis much more
constrained and the delicate analysis to study the possibility
of the existence of branched points is beyond the scope of this
chapter which is just intended to motivated the subsequent ones
on general conformally invariant variational problems. It has
beenprovedbyR.Ossermanthatuhasnointeriortrue branched
point (see [Oss]) and by R.Gulliver, R.Osserman and H.Royden
that u has no interior false branched point neither (see [GOR]).
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