Uniformisation of the twice–punctured disc - problems of confluence

Abstract

For the twice-punctured unit disc Ω p = { z : | z | < 1, z ≠ ± p }, where 0 < p < 1, we obtain precise descriptions for p near 0 of various parameters associated with the uniformisation of Ω p by the upper half-plane U = {τ: Im τ > 0}. These parameters include the hyperbolic length of the geodesic surrounding ± p , the so-called “accessory parameters”, and the “proximity parameter” which determines the behaviour of the hyperbolic density near the punctures of Ω p .