A characterization of domains in C 2 with noncompact automorphism group

Abstract

Abstract. Let D be a bounded domain in C 2 with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.

On the Automorphism Groups of Models in ℂ 2 $\mathbb {C}^{2}$

“…For the case that ∂Ω is real analytic and of D'Angelo finite type near a boundary orbit accumulation point (without the hypothesis of pseudoconvexity), a similar result as the above corollary was obtained in [29] by using a different method. In addition, it was shown in [3] that a smoothly bounded Ω in C 2 with real analytic boundary and with noncompact automorphism group, must be biholomorphically equivalent to E 1,m .…”

Characterizing domains by the limit set of their automorphism group

“…There are many other results characterizing special domains via the properties of their automorphism group and boundary, see for instance [GK87,Kim92,Won95,Gau97,Ver09] and the survey paper [IK99]. Like the two theorems mentioned above, almost all previous work assumes that either the entire boundary or a point in the limit set satisfies some infinitesimal condition (for instance strong pseudoconvexity, finite type, or Levi flat).…”

Model domains in ℂ3with abelian automorphism group

“…2 ) + µz 3 . For an automorphism g 0 of the polynomial domain Ω 0 ⊂ C 2 of this form, we have by the proof of proposition 2.7 of [17]…”

“…For instance, using this and an analysis of holomorphic tangent vector fields, it has been shown (see [2], [5]) that a bounded domain in C 2 , which is smooth weakly pseudoconvex and of finite type 2m, near a boundary orbit accumulation point must be equivalent to its model domain of the form (z 1 , z 2 ) ∈ C 2 : 2ℜz 2 + P (z 1 , z 1 ) < 0 where P (z 1 , z 1 ) is a homogeneous subharmonic polynomial of degree 2m without harmonic terms. The pseudoconvexity hypothesis on ∂D near the boundary orbit accumulation point was dropped in [3] and more recently in [17]. The Greene-Krantz conjecture, very well-known in this area states that a boundary orbit accumulation point must be of finite type.…”