Conformally Invariant Variational Problems. - SAM

This proves the theorem under the assumption that u(0) = 0.
Proof of the theorem in the general case. For a ∈ C and
| a |< 1 we define
v a (z) = z −a
1−āz
| v a (z) | 2 1 ⇐⇒ | z −a | 2 | 1−āz | 2
⇐⇒ (1− | a | 2 ) | z | 2 1− | a | 2
⇐⇒ | z | 2 1
Thus v a is a holomorphic map from D 2 into D 2 .
∀z ∈ D 2 v −a ◦v a (z) =
z−a
1−āz +a
1+ā z−a
1−āz
= z− | a |2 z
1− | a | 2 = z
This implies that v a is bijective and the inverse of v a is given by
v −a . Moreover the differential of v a nowhere degenerates, thus
v a ∈ M + (D 2 ). Observe that v −a (0) = a. Let u ∈ M + (D 2 )
and let a := u −1 (0)
u◦v −a (0) = 0
Applying the first part of the proof to u◦v −a (z) we deduce the
existence of θ ∈ R such that
u◦v −a (z) = e iO z
this implies (II.23) and theorem II.7 is proved.
✷
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