Uniformization of simply connected finite type log-Riemann surfaces

TL;DR

This paper characterizes simply connected finite type log-Riemann surfaces by showing they are biholomorphic to the complex plane and can be uniformized by specific entire functions involving polynomial-exponential integrals.

Contribution

It establishes a precise correspondence between these surfaces and a class of entire functions, providing a complete uniformization description.

Findings

Log-Riemann surfaces are biholomorphic to C.

Uniformizations are given by integrals of polynomial times exponential functions.

Any such entire function defines a corresponding log-Riemann surface.

Abstract

We consider simply connected log-Riemann surfaces with a finite number of ramification points. We prove that these surfaces are biholomorphic to C with uniformizations given by entire functions of the form F (z) = \int Q(z) e^{P(z)} dz where P, Q are polynomials of degrees equal to the number of infinite and finite order ramification points respectively. Conversely any such entire function defines a simply connected log-Riemann surface with finitely many ramification points.