sol108c6

sol108c6 - (1) Determine all conformal maps of D(0,...

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(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

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sol108c6 - (1) Determine all conformal maps of D(0,...

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