The Uniformization of a Hyperbolic Riemann Surface

In this paper a hyperbolic Riemann surface is described, and the uniformizing function for this surface is constructed. The uniformizing function for this hyperbolic surface is an infinite product similar to the ones constructed for some parabolic surfaces in earlier studies [1], [2], [4]. Description of the surface. Let I denote the positive integers. For each iEI, let C, be the vertical line segment in the Z-plane of length 2 centered at the point (1/t, 0). Let Si be the Z-sphere cut along G and for each iEI, i>l, let Si be the Z-sphere cut along C,_i and along d. For each iEI, join S, to Si+i along C; in such a way as to form first-order branch points over {l/i, 1) and il/i, — 1). The resulting Riemann surface F is simply connected and is easily seen to be hyperbolic because the surface has a free edge along the vertical line segment from — i to i. Approximating surfaces. For each iEI, let F{ be the Riemann surface formed from the first i sheets of F with the points of the curve d on Si deleted. Let Ff be the closed surface formed from the first i sheets of F with C, on 5, healed. There exists a unique function g,mapping Ff onto the Z-sphere such that fi=gl1, ftiO) =0ESi, fi (0) = 1, and fii oo ) = oo ESi. Let g be the unique function mapping F onto the disk {\z\<R^°°} such that if f=g~\ then /(0) =0£5i and/'(0) = l.