#### DOCUMENTA MATHEMATICA, Vol. 14 (2009), 673-689

Christian Miebach, Karl Oeljeklaus

On Proper 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.

2000 Mathematics Subject Classification: 32E10; 32M05; 32Q45; 32T05

Keywords and Phrases: Stein manifolds; bounded domains of holomorphy; proper actions; quotient by a discrete group

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