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 $\Omega$ such that $\mathrm{Aut}(\Omega )=G$. For Bedford–Dadok’s $\Omega$, $0\le {\dim }_C\Omega -{\dim }_{R}G\le 1$; for generic Saerens–Zame’s $\Omega$, ${\dim }_C\Omega \gg {\dim }_{R}G$.

J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds $\Omega$ such that $\mathrm{Aut}(\Omega )=G$. In his construction, it is typical that ${\dim }_C\Omega \gg {\dim }_{R}G$.

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 $\Omega$ such that $\mathrm{Aut}(\Omega )=G$ with ${\dim }_C\Omega ={\dim }_{R}G$. Working on a natural complexification of the real-analytic manifold $G$, our construction of $\Omega$ is geometrically concrete and elementary in nature.