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.
In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact $\overline\partial$-Neumann operator $N_q$ ($1\leq q\leq n-1$)? We prove that a smooth bounded pseudoconvex domain $Ω\subseteq\mathbb{C}^3$ with a one-dimensional complex manifold $M$ in its boundary has a noncompact Neumann operator on $(0,1)$-forms, under the additional assumption that $bΩ$ has finite regular D'Angelo $2$-type at a point of $M$, improving previous results of Fu, …
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.
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.
We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case…
Let $D$ be a bounded domain in $\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type.
We give an explicit description of hyperbolic Reinhardt domains D in C^2 such that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii) D intersects at least one of the coordinate complex lines $\{z_1=0\}$, $\{z_2=0\}$, and (iii) D has noncompact automorphism group. We also give an example that explains why such a setting is natural for the case of hyperbolic domains and an example that indicates that the situation in C^n for n greater than or equal to 3 is essentially more complicated than that in C^2.
The purpose of this article is twofold. The first aim is to characterize $h$-extendibility of smoothly bounded pseudoconvex domains in $\mathbb C^{n+1}$ 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 $\mathbb C^{n+1}$, then this point must be strongly pseudoconvex.
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 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 $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ 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 th…
The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater than 3, it has been pointed out by D. Burns that this result follows from known results on biholomorphism groups of complex manifolds with boundary and the fact that any such CR manifold M can be realized as the boundary of an analytic variety. When M is 3-dimensional, Burns's proof breaks down because abstract CR 3-manifolds are generically not realizable a…
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.
In this paper, we deal with a strongly pseudoconvex almost CR manifold with a CR contraction. We will prove that the stable manifold of the CR contaction is CR equivalent to the Heisenberg group model.
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.
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…
We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.
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.
We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint w…
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 $\mathbb C^2$.
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.
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.
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.
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…
In this paper, we study the existence of a complete holomorphic vector fields on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler-Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler-Einstein metric whose differential has a constant length. Then we will construct a complete holomorphic vector field from the gradient vector field of the potential function.
This paper deals with proper holomorphic self-maps of smoothly bounded pseudoconvex domains in $\C^2$. We study the dynamical properties of their extension to the boundary and show that their non-wandering sets are always contained in the weakly pseudoconvex part of the boundary. In the case of complete circular domains, we combine this fact with an entropy/degree argument to show that the maps are automorphisms. Some of our results remain true in $\C^n$.
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.
