The uniformization of a hyperbolic Riemann surface

Abstract

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 iEEI, let Ci be the vertical line segment in the Z-plane of length 2 centered at the point (1/i, 0). Let Si be the Z-sphere cut along Ci and for each iEI, i>1, let Si be the Z-sphere cut along Ci-i and along Ci, For each iEI, join Si to Si+, along Ci in such a way as to form first-order branch points over (1/i, 1) and (1/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 i( I, let Fi be the Riemann surface formed from the first i sheets of F with the points of the curve Ci on Si deleted. Let F* be the closed surface formed from the first i sheets of F with Ci on Si healed. There exists a unique function gi mapping F* onto the Z-sphere such that fi=g -, fi(O) =0 ESi, fi' (0) = 1, and fi( oo) = oo ESi. Let g be the unique function mapping F onto the disk | z <R < oo } such that if f=g-1, then f(O) =O E S and f'(O) = 1.