Complete Localization of Domains with Noncompact Automorphism Groups

Abstract

ABSTRACT. We prove a characterization of the domains in en with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.

Limits of an increasing sequence of complex manifolds

“…More information on this can be found in [14,29] for example. Recalling that p j is the image of z 0 under the biholomorphisim ψ j : D j → , we shall modify the version of Frankel's scaling technique in Kim's article [17] (cf. also [18]) to apply it to our situation.…”

Analysis of Orbit Accumulation Points and The Greene–Krantz Conjecture

“…Thus, it is enough to study the limit of ∂Ω j . Looking at Equation (4), by the existence of ∂ Ω (otherwise, by [8] Ω is not bounded), the right hand side must converge to a function, as the left hand side already converges to a function. Otherwise, the limit domain will be v > ∞ or v > 0, neither of which is possible for a bounded domain.…”

“…converge to nonzero constants (see [8]). At the same time, ρ k converges to a function of (z, w), while the higher terms converge to 0.…”

“…When a bounded convex domain Ω in C 2 admits a noncompact automorphism group, Frankel was able to construct in [5] a new (in general unbounded) domain Ω ′ which is biholomorphic, by the inverse of the limit of (Jφ j (q)) −1 (φ j (z, w) − φ j (q)), to Ω, where q ∈ Ω is an arbitrary interior point. Soon after that, Kim described the domain Ω ′ in [8] by the defining function. His argument is based on the fact that the boundary of Ω is invariant under automorphism and that the boundary of Ω ′ must exist (otherwise, Ω ′ can not be hyperbolic, i.e., Ω ′ can not be equivalent to a bounded domain in C 2 ).…”

“…

In C 2 , we classify the domains for which Aut(Ω) is noncompact and describe these domains by their defining functions. This note is based on the technique of the scaling method introduced by Frankel [5] and Kim [8]. One feature of this article is that we are able to analyze the defining functions of infinite type boundary.

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