Uniformization of higher genus finite type log-Riemann surfaces

TL;DR

This paper proves that higher genus finite type log-Riemann surfaces can be uniformized as punctured Riemann surfaces with exponential-type singularities in the pulled-back differential forms.

Contribution

It establishes a uniformization theorem for higher genus finite type log-Riemann surfaces, characterizing their structure via biholomorphism to punctured Riemann surfaces with exponential singularities.

Findings

Log-Riemann surfaces are biholomorphic to punctured Riemann surfaces.

The pull-back of the differential has exponential-type singularities.

The structure generalizes classical uniformization results to higher genus cases.

Abstract

We consider a log-Riemann surface $\mathcal{S}$ with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism $\pi : \mathcal{S} \to \C$. We prove that $\mathcal{S}$ is biholomorphic to a compact Riemann surface with finitely many punctures $S$, and the pull-back of the 1-form $d\pi$ under the biholomorphic map $\phi : S \to \mathcal{S}$ is a 1-form $\omega = \phi^* d\pi$ with isolated singularities at the punctures of exponential type, i.e. near each puncture $p$, $\omega = e^h \cdot \omega_0$ where $h$ is a function meromorphic near $p$ and $\omega_0$ a 1-form meromorphic near $p$.