Uniformization of a Once-Punctured Annulus

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 uniformization 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 degenerate cases are also treated.

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

“…7]. For example, it would be interesting to perform a detailed study of cosmological trajectories for some triply-connected non-elementary planar surfaces such as the twice-punctured disk [36][37][38][39] and once-punctured annulus [40].…”