Uniformization of simply connected finite type Log-Riemann surfaces

Abstract

We consider a log-Riemann surface S with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism π : S → C. We prove that S is biholomorphic to a compact Riemann surface with finitely many punctures S, and the pull-back of the 1-form dπ under the biholomorphic map φ : S → S is a 1-form ω = φ * dπ with isolated singularities at the punctures of exponential type, i.e. near each puncture p, ω = e h · ω 0 where h is a function meromorphic near p and ω 0 a 1-form meromorphic near p.

A Torelli type theorem for exp-algebraic curves

“…In [4], it is shown that a log-Riemann surface of finite type (which has at least one infinite order ramification point) is of the form (S , π), where S is a punctured compact Riemann surface S = S − {p 1 , . .…”

“…, p n . As described in [4], each puncture p j corresponds to an end of the log-Riemann surface where d j infinite order ramification points are added, d j being the order of the pole of h j at p j . Let w * be an infinite order ramification point associated to a puncture p j .…”

“…We will only consider those for which the set of infinite order ramification points is nonempty (otherwise the map π has finite degree and is given by a meromorphic function on a compact Riemann surface). In [4,7], uniformization theorems were proved for log-Riemann surfaces of finite type, which imply that a log-Riemann surface of finite type is given by a pair (S = S − {p 1 , . .…”

“…. Log-Riemann surfaces of finite type and exp-algebraic curvesWe recall some basic definitions and facts from[4,6,7].…”

The Ramificant Determinant

“…We study this problem in the simplest situation of genus 0, i.e., we assume that S × is simply connected. Then S × is parabolic and biholomorphic to C (see [4,5]). Also we proved there (see also the early work by R. Nevanlinna [13,14] and M. Taniguchi [18,19]) that we have an explicit formula for the uniformizationF : C → S × that is given by an entire function F = π •F of the form F (z) = Q(z)e P (z) dz, (1.1) where P and Q are polynomials of respective degrees d 1 and d 2 , where d 1 , resp.…”