The Accessory Parameter Problem for the Uniformization of the Twice‐Punctured Disc

Abstract

For the twice-punctured unit disc Ωp = {|z| < 1 : z ≠ ±p), where 0 < p < 1, a study is made of the socalled ‘accessory parameter problem’ which is associated with the uniformization of Ωp by the upper halfplane U = {Im τ > 0}. In particular, if the accessory parameter at p is denoted by m = m{p), then by describing m in terms of a variational problem, and by using an infinite continued fraction, m is shown to be a strictly increasing function of p, upper and lower bounds for m are given, and a precise picture of m asp→1 is obtained.