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Commentarii Mathematici Helvetici

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Volume 82, Issue 2, 2007, pp. 371–383
DOI: 10.4171/CMH/95

Published online: 2007-06-30

Complete hyperbolic Stein manifolds with prescribed automorphism groups

Su-Jen Kan[1]

(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 ≤ dimΩ − dimG ≤ 1; for generic Saerens–Zame’s Ω, dimΩ ≫ dimG.
  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 dimΩ ≫ dimG.
  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 dimΩ = dimG. 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