Domains in $\cx {n+1}$ with Noncompact Automorphism Group. II

On the automorphism groups of models in $\mathbb C^2$

In this note, we consider models in $\mathbb C^2$. The purpose of this note is twofold. We first show 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 $\mathbb C^2$.

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

We study and generalize a classical theorem of 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.

A Class of Domains with noncompact $\bar{\partial}$-Neumann operator

The $\bar{\partial}$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle, or have such a generalized Hartogs triangle imbedded appropriately in them.

Complete hyperbolic Stein manifolds with prescribed automorphism groups

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 could be prescribed asthe automorphism group of certain complex manifold. Whenthe Lie group $G$ is compact and connected, this problem has been completelysolved by Bedford-Dadok and independently by Saerens-Zame on 1987. Theyhave constructed \spc bounded domains $Ω$ such that $Aut(Ω)=G$. For Bedford-Dadok's $Ω, 0\le dim_{\Bbb C}Ω- dim_{\Bbb R}G\le 1$; for generic Saerens-Zame's$Ω,dim_{\Bbb C}Ω\gg dim_{\Bbb R}G$.J. Winkelmann has answered affirma…

A family of adapted complexifications for noncompact semisimple Lie groups

The maximal complexifications adapted to the Levi Civita connection for a distinguished one-parameter family of left-invariant metrics on a real semisimple Lie group G are determined. For G = SL(2,R) their realizations as invariant Riemann domains into SL(2,C) are carried out and their complex-geometric properties are investigated.

The Automorphism Group of a Domain with an Exponentially Flat Boundary Point

We study the automorphisms group action on a bounded domain in $\CC^n$ having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.

On the Dimensions of the Automorphism Groups of Hyperbolic Reinhardt Domains

We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.

Model Domains in C^3 with abelian automorphism group

It is shown that every hyperbolic rigid polynomial domain in C^3 of finite type, with abelian automorphism group is equivalent to a domain that is balanaced with respect to some weight.

Finitely smooth Reinhardt domains with non-compact automorphism group

We give a complete description of bounded Reinhardt domains of finite boundary smoothness that have non-compact automorphism group. As part of this program, we show that the classification of domains with non-compact automorphism group and having only finite boundary smoothness is considerably more complicated than the classification of such domains that have infinitely smooth boundary.

Automorphism groups of domains that depend on fewer than the maximal number of parameters

We study domains in complex $n$-space with automorphism group that does not depend on the full $n$ dimensions of the ambient space. A sufficient geometric condition is obtained to guarantee that a domain has such a "thin" automorphism group. Examples are provided to illustrate the ideas.

An Example of a Domain with Non-Compact Automorphism Group

We give an example of a bounded, pseudoconvex, circular domain in ${\mathbb C}^3$ with smooth, real-analytic boundary and non-compact automorphism group, which is not biholomorphically equivalent to any Reinhardt domain.

Perturbation of domains and automorphism groups

The paper is devoted to the description of changes of the structure of the holomorphic automorphism group of a bounded domain in ${\Bbb C}^n$ under small perturbation of this domain in the Hausdorff metric. We consider a number of examples when an arbitrary small perturbation can lead to a domain with a larger group, present theorems concerning upper semicontinuity property of some invariants of automorphism groups. We also prove that the dimension of an abelian subgroup of the automorphism group of a bounded domain in ${\Bbb C}^n$ does not exceed $n$.

Domains in complex surfaces with a non compact automorphism group - II

Let $X$ be an arbitrary complex surface and $D \subset X$ a domain that has a non compact group of holomorphic automorphisms. A characterization of those domains $D$ that admit a smooth real analytic, finite type boundary orbit accumulation point and whose closures are contained in a complete hyperbolic domain $D'\subset X$ is obtained.

On the degree of automorphisms of quasi-circular domains fixing the origin

By using the Bergman representative coordinates, we give the necessary and sufficient condition for the degree of automorphisms of quasi-circular domains fixing the origin to be equal to the resonance order, thus solving a conjecture of the author.

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in $\mathbb C^n$ in terms of the boundary behavior of automorphism orbits by using the scaling method.

On automorphisms of quasi-circular domains fixing the origin

It is known that automorphisms of quasi-circular domains fixing the origin are polynomial mappings. By introducing the so-called resonance order and quasi-resonance order, we provide a uniform upper bound for the degree of such polynomial automorphisms. As a particular consequence of our result, we obtain a generalization of the classical Cartan's theorem for circular domains to the quasi-circular case.

Domains with Non-Compact Automorphism Group: A Survey

We survey results arising from the study of domains in C^n with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points. Particular attention will be paid to results derived in the past ten years.

The automorphism group of a certain unbounded non-hyperbolic domain

In this paper we determine the automorphism group of the Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^n\times\mathbb{C}^m$ which is defined by the inequality ${\|ζ\|}^2<e^{-μ{\|z\|}^2}$.

蠕虫域的自同构

Automorphisms of the Worm Domain

本文针对多复变中作为关键反例的Diederich-Fornæss蠕虫域开展研究，考察其自同构群，明确了该域边界除例外轨迹与端帽外处处局部球形的性质。

On a domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group

In this paper, we prove that if D is a simply-connected domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group, then D is biholomorphic to the bidisc. The proof is based on a careful analysis of invariant measures.

Examples of domains with non-compact automorphism groups

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.

Semi-continuity of Automorphism Groups of Strongly Pseudoconvex Domains: the low differentiability case

We study the semicontinuity of automorphism groups for perturbations of domains in complex space or in complex manifolds. We provide a new approach to the study of such results for domains having minimal boundary smoothness. The emphasis in this study is on the low differentiability assumption and the new methodology developed accordingly.

Upper semicontinuity of the dimensions of automorphism groups of domains in $C^n$

Let $H^n$ be the metric space of all bounded domains in $C^n$ with the metric equal to the Hausdorff distance between boundaries of domains. We prove that the dimension of the group of automorphisms of domains is an upper semicontinuous function on $\H^n$. We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in $H^n$.

Semicontinuity of the Automorphism Groups of Domains with Rough Boundary

Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in $\CC^n$, $n > 1$, with Lipschitz boundary, but it holds for domains in $\CC^1$ with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings.

On the automorphism group of a certain infinite type domain in $\mathbb C^2$

In this article, we consider an infinite type domain $Ω_P$ in $\mathbb C^2$. The purpose of this paper is to investigate the holomorphic vector fields tangent to an infinite type model in $\mathbb C^2$ vanishing at an infinite type point and to give an explicit description of the automorphism group of $Ω_P$.

On the linearity of origin-preserving automorphisms of quasi-circular domains in $\mathbb C^n$

A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in $\mathbb C^n$. Our main result is applied to obtain some simple criterions for the case $n=3$ and to prove that Braun-Kaup-Upmeier's theorem remains true for our class of quasi-circular domains.

将半单李群实现为有界域的全纯自同构群

Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains

本文证明了包括所有连通实半单群在内的一大类李群（含许多非线性李群）均可实现为仿射复空间有界域的全纯自同构群，推广了1990年关于连通线性李群的结论。

On the boundary orbit accumulation set for a domain with non-compact automorphism group

For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is a compact subset of $\partial D$ and, if $S(D)$ contains at least three points, it is connected and thus has the power of the continuum. We also discuss how $S(D)$ relates to other invariant subsets of $\partial D$ and show in particular that $S(D)$ is always a subset of the Šilov boundary.

在复仿射空间$\mathbb C^n$中Runge域之间biholomorphic映射的近似

Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms

本文研究了复仿射空间中Runge域之间biholomorphic映射的近似问题，证明其可由全纯自同构逼近，并推广了相关结果。

Cycle Connectivity and Automorphism Groups of Flag Domains

A flag domain $D$ is an open orbit of a real form $G_0$ in a flag manifold $Z=G/P$ of its complexification. If $D$ is holomorphically convex, then, since it is a product of a Hermitian symmetric space of bounded type and a compact flag manifold, ${Aut}(D)$ is easily described. If $D$ is not holomorphically convex, then in our previous work (American J. Math, 136, Nr.2 (2013) 291-310 (arXiv: 1003.5974)) it was shown that ${Aut}(D)$ is a Lie group whose connected component at the identity agrees with $G_0$ except possibly in situations which arise in Onishchik's list of flag manifolds where ${Au…

Fatou-Bieberbach domains as basins of attraction of automorphisms tangent to the identity

We prove that there exists an automorphism of C^2 tangent to the identity with a domain of attraction to the origin, biholomorphic to the origin, along a degenerate characteristic direction.

The automorphism group of the tetrablock

The tetrablock is a domain in 3-dimensional complex space that meets 3-dimensional Euclidean space in a regular tetrahedron. It is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation of the tetrablock by complex geodesic discs.

NC Automorphisms of nc-bounded domains

We establish rigidity (or uniqueness) theorems for nc automorphisms which are natural extensions of clasical results of H.~Cartan and are improvements of recent results. We apply our results to nc-domains consisting of unit balls of rectangular matrices.

Lacunary series, resonances, and automorphisms of $\mathbb{C}^2$ with a round Siegel domain

We construct transcendental automorphims of $\mathbb{C}^{2}$ having an unbounded and regular Siegel domain.

No bounded geometry wandering domains for sufficiently regular automorphisms

A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of $n$-tori, $n\ge2$, and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.

Automorphisms of $\mathbb{C}^m$ with bounded wandering domains

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.

Automorphisms of normal quasi-circular domains

It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal quasi-circular domains in $\mathbb C^2$ is linear.

The magic functions and automorphisms of a domain

We introduce the notion of magic functions of a general domain in d-dimensional complex space and show that the set of magic functions of a given domain is an intrinsic complex-geometric object. We determine the set of magic functions of the symmetrised bidisc G, and thereby find all automorphisms of G and a formula for the Caratheodory distance on G.

Rigidity of Proper Holomorphic Self-mappings of the Pentablock

The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.

The automorphism group and limit set of a bounded domain II: the convex case

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non-compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the…

可实现为有界域自同构群的非线性李群

Non-linear Lie groups that can be realized as automorphism groups of bounded domains

本文研究李群能否成为 C^n 有界域双全纯自同构群，在1990年连通线性李群结论基础上，给出满足该实现性质的非线性李群例子。

III型拟对称Siegel域的全纯自同构群的相干态表示

Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III

本文研究了III型拟对称Siegel域全纯自同构群的泛相干态表示分类，揭示了其与Kähler代数结构理论的联系，并提出基于广义Iwasawa分解的方法。

The automorphism group and limit set of a bounded domain I: the finite type case

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to ${ \rm Aut}(\mathbb{B}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of th…

Dynamics of Rational Surface Automorphisms: Rotation Domains

We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.

Characterizing domains by the limit set of their automorphism group

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene-Krantz conjecture in the case of uniform non-tangential…

On the automorphism group of certain Short $\mathbb C^2$'s

For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not= 0$, it is known that the sub-level sets of the Green's function $G^+_H$ associated with $H$ are Short $\mathbb C^2$'s. For a given $c > 0$, we study the holomorphic automorphism group of such a Short $\mathbb C^2$, namely $Ω_c = \{ G^+_H < c \}$. The unbounded domain $Ω_c \subset \mathbb C^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $Ω_c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism grou…

Characterization of the Hilbert ball by its automorphism group

Let $Ω$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families'' arguments in the construction of the limit biholomorphism.

A Kobayashi metric version of Bun Wong's theorem

We prove that a strongly pseudoconvex domain with noncompact group of Kobayashi/Royden metric isometries must be biholomorphic to the unit ball.

On a characterization of unbounded homogeneous domains with boundaries of light cone type

We determine the automorphism groups of unbounded homogeneous domains with boundaries of light cone type. Furthermore we present the group-theoretic characterization of the domain. As a corollary we prove the non-existence of compact quotients of a unbounded homogeneous domain. We also give a counterexample of the characterization.

Hyperbolicity of cycle spaces and automorphism groups of flag domains

If G_0 is a real form of a complex semisimple Lie group G and Z is compact G-homogeneous projective algebraic manifold, then G_0 has only finitely many orbits on Z. Complex analytic properties of open G_0-orbits D (flag domains) are studied. Schubert incidence-geometry is used to prove the Kobayashi hyperbolicity of certain cycle space components C_q(D). Using the hyperbolicity of C_q(D) and analyzing the action of Aut(D) on it, an exact description of Aut(D) is given. It is shown that, except in the easily understood case where D is holomorphically convex with a nontrivial Remmert reduction, …

Quotients of bounded homogeneous domains by cyclic groups

Let $D$ be a bounded homogeneous domain in $\mbb{C}^n$ and let $Γ$ be a cyclic discrete subgroup of the automorphism group of $D$. It is shown that the complex space $D/Γ$ is Stein.

On the automorphism groups of algebraic bounded domains

Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group $Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a real Lie group such that the action $Aut(D)\times D\to D$ is real analytic. This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here we restrict our attention to the class of domains $D$ defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we conside…

Stably-interior points and the Semicontinuity of the Automorphism group

The first result is the semicontinuity of automorphism groups for the collection of complex two-dimensional bounded pseudoconvex domains with smooth boundary of finite D'Angelo type. The method of proof is new so that it simplifies the previous proof of earlier semicontinuity theorems on bounded strongly pseudoconvex daomains by Greene and Krantz in the early 1980s.

Rigidity results for automorphisms of Hardy-Toeplitz $C^*$-algebras

We prove a number of results on the automorphisms of and isomorphisms between Hardy-Toeplitz algebras $\mathcal{T}(D)$ associated to bounded symmetric domains $D$: that the stable isomorphism class of $\mathcal{T}(D)$ determines $D$ (even when it is reducible), that for reducible domains $D=D_1\times\cdots \times D_s$ the automorphisms of the Shilov boundary $\check{S}(D)$ induced by those of $\mathcal{T}(D)$ permute the Shilov boundaries $\check{S}(D_i)$, and that by contrast to arbitrary solvable algebras, automorphisms of $\mathcal{T}(D)$ that are trivial on their character spaces $\check{S…