Complete hyperbolic Stein manifolds with prescribed automorphism groups

It is well-known that the automorphism group of a hyperbolic manifold is a Lie group.Conversely, it is interesting to see whether or not any Lie group could be prescribed asthe automorphism group of certain complex manifold. Whenthe Lie group $G$ is compact and connected, this problem has been completelysolved by Bedford-Dadok and independently by Saerens-Zame on 1987. Theyhave constructed \spc bounded domains $\Omega$ such that $Aut(\Omega)=G$. For Bedford-Dadok's $\Omega, 0\le dim_{\Bbb C}\Omega- dim_{\Bbb R}G\le 1$; for generic Saerens-Zame's$\Omega,dim_{\Bbb C}\Omega \gg dim_{\Bbb R}G$.J. Winkelmann has answered affirmatively to noncompact connected Liegroups in recent years. He showed there exist Stein complete hyperbolic manifolds $\Omega$ such that $Aut(\Omega)=G$.In his construction, it is typical that $dim_{\Bbb C}\Omega\gg dim_{\Bbb R}G$.In this article, we tackle this problem from a different aspect. We provethat for any connected Lie group $G$ (compact or noncompact), there exist completehyperbolic Stein manifolds $\Omega$ such that $Aut(\Omega)=G$ with $dim_{\BbbC}\Omega=dim_{\Bbb R}G.$ Working on a natural complexification of the real-analyticmanifold $G$, our construction of $\Omega$ is geometrically concrete andelementary in nature.