Abstract. The explicit formula for the hyperbolic metric λ α, β, γ (z)|dz| on the thrice-punctured sphere P\{0, 1, ∞} with singularities of order 0 < α, β < 1, γ ≤ 1, α + β + γ > 2 at 0, 1, ∞ was given by Kraus, Roth and Sugawa in [9]. In this article we investigate the asymptotic properties of the higher order derivatives of λ α, β, γ (z) near the origin and give more precise descriptions for the asymptotic behavior of λ α, β, γ (z).
We define a distance function on the bordered punctured disk 0 < |z| ≤ 1/e in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk 0 < |z| < 1. As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.
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.
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