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

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.