On proper $\mbb{R}$-actions on hyperbolic Stein surfaces

TL;DR

This paper studies proper real-line actions on hyperbolic Stein surfaces, showing that certain quotients are Stein manifolds and deriving a normal form for the domain, advancing understanding of complex geometric actions.

Contribution

It proves that quotients of certain hyperbolic Stein domains by proper $br$-actions are Stein and provides a normal form for these domains, which was previously unknown.

Findings

Quotients of hyperbolic Stein domains by proper $br$-actions are Stein manifolds.

A normal form for such domains is established.

The results enhance understanding of $br$-actions on complex surfaces.

Abstract

In this paper we investigate proper $\mbb{R}$--actions on hyperbolic Stein surfaces and prove in particular the following result: Let $D\subset\mbb{C}^2$ be a simply-connected bounded domain of holomorphy which admits a proper $\mbb{R}$--action by holomorphic transformations. The quotient $D/\mbb{Z}$ with respect to the induced proper $\mbb{Z}$--action is a Stein manifold. A normal form for the domain $D$ is deduced.