[Paper Review] Examples of domains with non-compact automorphism groups

Siqi Fu, Alexander Isaev+1 more|arXiv (Cornell University)|Feb 9, 1996

Holomorphic and Operator Theory5 references3 citations

TL;DR

This paper constructs bounded, pseudoconvex, circular domains in complex dimensions three and higher with smooth real analytic boundaries and non-compact automorphism groups that are not biholomorphically equivalent to any Reinhardt domain. In dimension two, a similar example is given, but the domain has a non-Lipschitz boundary point, demonstrating that smoothness is essential for equivalence to Reinhardt domains in higher dimensions.

ABSTRACT

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.

Motivation & Objective

• To construct bounded, pseudoconvex, circular domains in C^n (n ≥ 3) with non-compact automorphism groups.
• To show such domains are not biholomorphically equivalent to any Reinhardt domain, challenging the expectation that non-compact automorphism groups imply Reinhardt structure.
• To explore the role of boundary regularity by constructing a dimension-two example with a non-Lipschitz boundary point.
• To clarify the geometric and analytic conditions under which non-compact automorphism groups imply Reinhardt symmetry.
• To extend understanding of automorphism group behavior in pseudoconvex domains beyond known classes like Reinhardt domains.

Proposed method

• Constructing a bounded, pseudoconvex, circular domain in C^n (n ≥ 3) with smooth real analytic boundary.
• Using the circular symmetry and pseudoconvexity to analyze the automorphism group structure.
• Applying biholomorphic invariants and boundary regularity criteria to rule out equivalence to Reinhardt domains.
• Employing a limiting construction in dimension two to produce a domain with a non-Lipschitz boundary point.
• Analyzing the automorphism group's non-compactness via group action on the boundary and curvature properties.
• Using differential geometric and complex analytic tools to verify smoothness and non-equivalence to Reinhardt domains.

Experimental results

Research questions

• RQ1 Can a bounded, pseudoconvex, circular domain in C^n (n ≥ 3) have a non-compact automorphism group without being biholomorphically equivalent to a Reinhardt domain?
• RQ2 What role does boundary regularity play in determining whether a domain with non-compact automorphism group must be Reinhardt?
• RQ3 Is it possible to construct such a domain in dimension two with a non-smooth boundary point while preserving non-compact automorphism group?
• RQ4 How do the geometric and analytic properties of the boundary influence the structure of the automorphism group?
• RQ5 Are there intrinsic invariants that distinguish non-Reinhardt domains with non-compact automorphism groups from Reinhardt domains?

Key findings

• The authors construct a bounded, pseudoconvex, circular domain in C^n (n ≥ 3) with smooth real analytic boundary and non-compact automorphism group that is not biholomorphically equivalent to any Reinhardt domain.
• In dimension two, a similar domain is constructed, but it fails to be Lipschitz regular at one boundary point, indicating that smoothness is a necessary condition for such equivalences.
• The non-compactness of the automorphism group is preserved in both constructions, demonstrating that such groups do not necessarily imply Reinhardt structure.
• The boundary regularity condition—specifically, C^∞ or real analytic smoothness—is essential for the equivalence to Reinhardt domains.
• The results show that the class of domains with non-compact automorphism groups is strictly larger than the class of Reinhardt domains when boundary smoothness is not assumed.
• The examples provide counterexamples to the conjecture that all bounded domains with non-compact automorphism groups and smooth boundaries must be Reinhardt.