Uniformization of a Once-Punctured Annulus
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Uniformization of a Once-Punctured Annulus Tanran Zhang
Received: 30 January 2014 / Revised: 31 July 2014 / Accepted: 24 August 2014 © Springer-Verlag Berlin Heidelberg 2014
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. Keywords Uniformization · Hyperbolic metric · Punctured annulus · Collar · Extremal length Mathematics Subject Classification
30F10 · 14Q05
1 Introduction The uniformization theorem implies that every Riemann surface X is conformally equivalent to the quotient space H/G, where G is a torsion-free Fuchsian group acting on the upper half-plane H := {z ∈ C : Imz > 0}, if X is not conformally equivalent to ˆ the complex plane C, the once-punctured complex plane C\{a} the Riemann sphere C,
Communicated by Mario Bonk. T. Zhang (B) Department of Mathematics, Soochow University, No.1 Shizi Street, Suzhou 215006, China e-mail: [email protected] T. Zhang Division of Mathematics, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
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or a complex torus. It is, however, difficult to find an explicit form of the holomorphic universal cover π or the covering group G, except for several special cases (see e.g., [8,17]). For a twice-punctured unit disk, Hempel and Smith [9–11] considered the uniformization problem and the hyperbolic metric, and Beardon [4] provided five parameters to characterize the twice-punctured disk via its hyperbolic structure and complex structure. Nevanlinna [14, I.3, I.4] introduced a method to regard the puncture as the extremal case when a boundary curve shrinks to a single point. In this article, we give five parameter pairs to uniformize a once-punctured annulus A. These parameter pairs can be divided into two classes corresponding to the hyperbolic structure and complex structure of A, respectively. Let γ be a simple closed geodesic on a hyperbolic surface X , and let Cθ (γ ) := {x ∈ X : δ X (x, γ ) < sinh−1 (tan θ )},
(1.1)
where δ X is the hyperbolic distance on X induced by the hyperbolic metric of the Gaussian curvature −1. (Some authors prefer the hyperbolic metric of the Gaussian curvature −4. For this reason one should be cautious when referring to other books or papers.) Cθ (γ ) is called a collar about γ of angular width θ if it is doubly connected. When X \γ has a doubly connected component W and γ is homotopic to no puncture, γ is homotopic to a border of X , and γ is called peri pheral. If Cθ (γ ) is a collar, θ (γ ) := Cθ (γ ) ∪ W is a doubly connected subdomain in X con
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