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## Classification Of Annuli

Differential Geometry
Tsakanikas Nickos
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### Classification Of Annuli

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 $A_{R} = \left\{ \, z \in \mathbb{C} \ \big| \ 1 < |z| < R \, \right\}$