Commentarii Mathematici Helvetici
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Published online: 2007-06-30
Complete hyperbolic Stein manifolds with prescribed automorphism groupsSu-Jen Kan (1) Academia Sinica, Taipei, Taiwan
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.
Keywords: Hyperbolic manifolds, Stein manifolds, automorphism groups
Kan Su-Jen: Complete hyperbolic Stein manifolds with prescribed automorphism groups. Comment. Math. Helv. 82 (2007), 371-383. doi: 10.4171/CMH/95