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Classification Of Annuli

Differential Geometry
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Tsakanikas Nickos
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Classification Of Annuli

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Post by Tsakanikas Nickos » Sat Oct 08, 2016 12:18 pm

Definition: An annulus is a Riemann surface $A$ with fundamental group $ \pi_{1}(A) \cong \mathbb{Z} $.

Show that an annulus $A$ is biholomorphic to one of the following Riemann surfaces:
  • the punctured disc $ \mathbb{D}^{*} $
  • the punctured plane $ \mathbb{C}^{*} $
  • a round annulus $ A_{R} = \left\{ \, z \in \mathbb{C} \ \big| \ 1 < |z| < R \, \right\} $
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