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

Abstract

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