Uniformization of Riemann Surface and their automorphisms

\mathbf{Thm1:} Every simply connected Riemann surface must comformally equivalent to one of the followings:

  • The Riemann sphere, \hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}
  • The complex plane, \mathbb{C}
  • The unit disk. \mathbb{D}=\{z\in \mathbb{C}||z|<1\}

\mathbf{Thm2:} Every Riemann surface M is conformally equivalent to \Sigma/G where \Sigma is one of standard types, \hat{\mathbb{C}}, \mathbb{C}, \mathbb{D}. G is a subgroup of \text{Aut}(\Sigma) that acts freely discontinuously on \Sigma. Furthermore G\cong \pi_1(M).

\mathbf{Thm3:} Two Riemann surfaces are conformally equivalent if and only if they have the same \Sigma and their G are conjugate in \text{Aut}(\Sigma).

Basic facts:

\text{Aut}(\hat{\mathbb{C}})= Mobius  transformations=PSL(2, \mathbb{C})= SL(2,\mathbb{C})/{\pm I}

\text{Aut}({\mathbb{C}})=\{az+b|a, b\in \mathbb{C}, a\neq 0\}=\text{Aff}(1,\mathbb{C})

\text{Aut}({\mathbb{D}})\cong PSL(2,\mathbb{R})

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