Non-compact Automorphism Group Research Articles

On the Manifolds with Noncompact Automorphism Groups

On the Manifolds with Noncompact Automorphism Groups

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in [Formula: see text] in terms of the boundary behavior of automorphism orbits by using the scaling method.

A method of potential scaling in the study of pseudoconvex domains with noncompact automorphism group

The affine scaling method has been a typical approach to study complex domains with noncompact automorphism group. In this article, we will introduce an alternative approach, so called, the method of potential scaling to construct a certain class of potential functions of the Kähler-Einstein metric. We will also prove that if a bounded pseudoconvex domain admits a potential function of the Kähler-Einstein metric whose differential has constant length, then there is an 1-parameter family of automorphisms.

Some properties of h-extendible domains in [formula omitted

The purpose of this article is twofold. The first aim is to characterize h-extendibility of smoothly bounded pseudoconvex domains in Cn+1 by their noncompact automorphism groups. Our second goal is to show that if the squeezing function tends to 1 at an h-extendible boundary point of a smooth pseudoconvex domain in Cn+1, then this point must be strongly pseudoconvex.

Automorphism group actions in complex analysis

In this paper we describe the subject of automorphism groups of domains in complex space. This has been an active area of research for fifty years or more, and continues to be dynamic and developing today. We discuss noncompact automorphism groups, the Bun Wong/Rosay theorem, the Greene/Krantz conjecture, semicontinuity of automorphism groups, the method of scaling, and other current topics. Contributions from geometers, Lie theorists, analysts, and function theorists are described.

Generic analytic polyhedron with non-compact automorphism group

In this paper we prove the following rigidity theorem: a generic analytic polyhedron with non-compact automorphism group is biholomorphic to the product of a complex manifold with compact automorphism group and a polydisk. Moreover, this complex manifold and the dimension of this polydisk can be explicitly described in terms of the limit set of the automorphism group.

On the domains with noncompact automorphism groups

For automorphism groups, we prove that the orbit accumulation points are either hyperbolic or parabolic. This answers the question pertaining to the non-compactness of the inverses of a non-compact sequence of automorphisms. We also partially extend a previous theorem of the author on global pseudoconvexity to higher dimensions, provided it admits a non-compact automorphism group. In general, a full extension does not exist. We provide an example to elaborate on the nonexistence. This example is carefully constructed and an exact sequence of automorphisms is explicitly found.

Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel

In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given.

Characterization of convex domains in ℂ2 with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ℂn. Suppose there exists {g j} ⊂ Aut(D) such that {g j(z)} accumulates at a boundary point p ∈ ∂D for some z ∈ D. Then ∂D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ℂ2 with C 2 boundary. Suppose that there is a sequence {g j} ⊂ Aut(D) such that {g j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ∂D is such an orbit accumulation point, ∂D contains no non-trivial analytic variety passing through p.

On h-extendible domains and associated models

A boundary point of a smooth pseudoconvexdomain in Cn is said to be h-extendible if its Catlin's multi-type coincides with its D'Angelo's multi-type. There is a local model defined by Catlin's multi-weight. In this paper, we show that a domain in Cn with a noncompact automorphism group is biholomorphically equivalent to its associated model if there exists a sequence of automorphisms of the domain that has an orbit converging to an h-extendible boundary point non-tangentially in a cone region.

On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains

AbstractWe study and generalize a classical theoremof L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group

Invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group

We study invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group. The main result is that the corresponding Bergman and K\(\ddot{\mathrm{a}}\)hler–Einstein metrics are metrically equivalent. We also determine the comparisons among invariant metrics, including the Caratheodory and Kobayashi pseudo-metrics additionally.

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

In this article, we first give a characterization of models in $\mathbb {C}^{2}$ by their noncompact automorphism groups. Then we give an explicit description for automorphism groups of models in $\mathbb {C}^{2}$ .

Proper holomorphic maps between bounded symmetric domains revisited

We prove that a proper holomorphic map between two bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively.

AUTOMORPHISMS OF THE HARTOGS TYPE DOMAINS OVER CLASSICAL SYMMETRIC DOMAINS

In this paper we completely describe the automorphism group of the Hartogs type domains over bounded classical symmetric domains by using the ball characterization theorem about noncompact automorphism groups. As an application, we give plenty of examples of non-homogeneous domains in ℂn, for which the real dimension of the automorphism group is less than n2 - 1.

Extension of an analytic disc and domains in ℂ2with noncompact automorphism group

Extension of an analytic disc and domains in ℂ²with noncompact automorphism group

Some Aspects of the Kobayashi and Carathéodory Metrics on Pseudoconvex Domains

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carathéodory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C 2, and on convex finite type domains in C n using the scaling method. Applications include an alternate proof of the Wong–Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C 2 and convex finite type domains in C n in terms of Euclidean parameters. Second, a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C 1-isometries of the Kobayashi and Carathéodory metrics on a smoothly bounded strongly pseudoconvex domain.

Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in ℂ2

We give a characterization of non-hyperbolic pseudoconvex Reinhardt domains in ℂ2 for which the answer to the Serre problem is positive. Moreover, all non-hyperbolic pseudoconvex Reinhardt domains in ℂ2 with non-compact automorphism group are explicitly described.

A Kobayashi metric version of Bun Wong's theorem

We study the classical theorem of Bun Wong and Rosay about domains with non-compact automorphism group and strongly pseudoconvex orbit accumulation point. We formulate and prove a version of the result in the language of the Kobayashi metric

Characterization of Domains in ℂn by their Noncompact Automorphism Groups

In this paper, the characterization of domains in ℂn by their noncompact automorphism groups are given.