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 can be prescribed as the automorphism group of a certain complex manifold. When the Lie group G is compact and connected, this problem has been completely solved by Bedford–Dadok and independently by Saerens–Zame in 1987. They have constructed strictly pseudoconvex bounded domains such that Aut( ) = G. For Bedford–Dadok’s , 0 ≤ dimC − dimR G ≤ 1; for generic Saerens–Zame’s , dimC dimR G. J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds such that Aut( ) = G. In his construction, it is typical that dimC dimR G. In this article, we tackle this problem from a different aspect. We prove that for any connected Lie group G (compact or noncompact), there exist complete hyperbolic Stein manifolds such that Aut( ) = G with dimC = dimR G. Working on a natural complexification of the real-analytic manifold G, our construction of is geometrically concrete and elementary in nature. Mathematics Subject Classification (2000). 32C09, 32Q28, 32Q45, 53C24, 58D19.