On affine Riemann surfaces

TL;DR

This paper proves that the universal cover of certain affine Riemann surfaces in complex two-space is the Euclidean plane, revealing the structure of their orbit spaces under covering group actions.

Contribution

It establishes that the universal cover of a regular level set of a smooth complex function in ${f C}^2$ is ${f R}^2$, clarifying the topological structure of these affine Riemann surfaces.

Findings

Universal cover of the affine Riemann surface is ${f R}^2$

Orbit space of the covering group action is the original surface

Provides insight into the topology of affine Riemann surfaces

Abstract

We show that the universal covering space of a connected component of a regular level set of a smooth complex valued function on ${\mathbb{C}}^2$, which is a smooth affine Riemann surface, is ${\mathbb{R}}^2$. This implies that the orbit space of the action of the covering group on ${\mathbb{R}}^2$ is the original affine Riemann surface.