Uniformisation of a once-punctured annulus

Tanran Zhang

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arXiv · 1401.7786

Uniformisation of a once-punctured annulus

Tanran Zhang

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Abstract

The universal cover or the covering group of a hyperbolic Riemann surface $X$ is important but hard to express explicitly. It can be, however, detected by the uniformisation and a suitable description of $X$. Beardon proposed five different ways to describe twice-punctured disks using fundamental domain, hyperbolic length, collar and extremal length in 2012. We parameterize a once-punctured annulus $A$ in terms of five parameter pairs and give explicit formulas about the hyperbolic structure and the complex structure of $A$. Several degenerating cases are also treated.

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Tanran Zhang. 2014-01-30. Uniformisation of a once-punctured annulus. https://arxiv.org/abs/1401.7786

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