Complete hyperbolic Stein manifolds with prescribed automorphism groups

Abstract

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 Ω such that Aut(Ω) = G. For Bedford–Dadok’s Ω, 0 ≤ dimℂΩ − dimℝG ≤ 1; for generic Saerens–Zame’s Ω, dimℂΩ ≫ dimℝG.
J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds Ω such that Aut(Ω)  = G. In his construction, it is typical that dimℂΩ ≫ dimℝG.
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 Ω such that Aut(Ω) = G with dimℂΩ = dimℝG. Working on a natural complexification of the real-analytic manifold G, our construction of Ω is geometrically concrete and elementary in nature.