sol108c6

# sol108c6 - (1 Determine all conformal maps of D(0 1\cfw_0...

This preview shows pages 1–2. Sign up to view the full content.

(1) Determine all conformal maps of D (0 , 1) \{ 0 } → D (0 , 1) \{ 0 } . Solution: Note that a conformal map f from U = D (0 , 1) \{ 0 } → U has a singularity at 0. As the map is bounded, it is clearly bounded near 0 and thus a removable singularity. Thus f extends to a holomorphic map ˆ f : D (0 , 1) C . The image of D (0 , 1) under this map is an open set, and it is also contained in the closure U = D (0 , 1) of U . The image of 0 can’t be in ∂D (0 , 1), as then ˆ f ( D (0 , 1)) would not be open (another argument is that | ˆ f | can’t have a maximum in an internal point), so ˆ f (0) = 0. Thus conformal maps U U are the restrictions to U of conformal maps D (0 , 1) D (0 , 1) which are 0 at 0. By Schwarz’s lemma these are only the rotations. We conclude that the only conformal maps D (0 , 1) \{ 0 } → D (0 , 1) \{ 0 } are f ( z ) = ωz for some ω C with | ω | = 1. (2) Deﬁne the automorphism group Aut ( H ) = { f : H H | f holomorphic and bijective } . Here H = { z | = ( z ) > 0 } denotes the upper half plane. Now show that Aut ( H ) is isomorphic to PSL 2 ( R ) = { M = ± a b c d ² | a,b,c,d R ,ad - bc = 1 } / where M N if and only if N = M or N = - M . Finally show that Aut ( D (0 , 1)) is also isomorpic to PSL 2 ( R ). Solution: Recall the Cayley map ψ ( z ) = ( z - i ) / ( z + i ) which maps H conformally to D (0 , 1). In particular given a conformal map f : H H we ﬁnd that ψ f ψ - 1 is a conformal map of D (0 , 1) D (0 , 1). This creates an isomorphism between Aut ( H ) and Aut ( D (0 , 1)) (note the map is clearly invertible). In particular any conformal map in Aut ( H ) can be written as ψ - 1 g ψ for a conformal map in Aut ( D (0 , 1)). By the classiﬁcation of Aut ( D (0 , 1)) we know that g is a fractional linear transformation, and so is ψ . Thus all elements of Aut ( H ) are fractional linear transformations. Moreover we see that an element of Aut ( H ) must preserve H = R . Let us therefore consider the FLTs which preserve R . They must satisfy the following equation for all z R az + b cz + d = az + b cz + d = ¯ az

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 12/08/2010.

### Page1 / 4

sol108c6 - (1 Determine all conformal maps of D(0 1\cfw_0...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online