Domain In Complex Space Research Articles

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.

Some non-pseudoconvex domains with explicitly computable non-Hausdorff Dolbeault cohomology

We explicitly compute the Dolbeault cohomologies of certain domains in complex space generalizing the classical Hartogs figure. The cohomology groups are non-Hausdorff topological vector spaces, and it is possible to identify the reduced (Hausdorff) and the indiscrete part of the cohomology.

On positive definite distributions with compact support

We propose necessary and sufficient conditions for a distribution (generalized function) $f$ of several variables to be positive definite. For this purpose, certain analytic extensions of $f$ to tubular domains in complex space $\mathbb{C}^n$ are studied. The main result is given in terms of completely monotonic functions on convex cones in $\mathbb{R}^n$.

The automorphism groups of domains in complex space: a survey

We consider recent developments in the study of automorphism groups of domains in complex space. Particular attention is paid to results with a basis in geometry.

Semicontinuity 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.

On special forms of holomorphic automorphisms of domains in complex two-space

The authors study geometric conditions on a domain in complex space which will guarantee that the automorphism group of the domain will depend on fewer than the maximal number of variables. A number of useful examples are provided.

A new invariant metric and applications

We study a new construction of an invariant metric for compact subgroups of the automorphism group of a domain in complex space. Applications are provided.

On propagation of boundary continuity of holomorphic functions of several variables

We consider continuity properties of analytic functions in bounded domains in complex space, focusing on the following two questions:Question 1. Given a bounded domain G in complex affine n-space. For which subsets S of the boundary of G does the following hold? If f is a bounded function which is analytic in G and extends continuously to the union of G and S then f extends continously to the closure of G.Question 2. Given a domain G as above and a function f analytic in G and continuous in the closure of G. For which subsets S of the boundary is the rate of continuity of f on the closure of G determined already by its rate of continuity along S?Previously it was known that in some cases, the set S may be taken to be the Shilov boundary of the algebra of analytic functions on the domain which are continuous on the closure of the domain, but examples when this is not sufficient were also known.In Paper I we prove that for a bounded, smoothly bounded pseudoconvex domain G in 2-dimensional complex affine space one may take for S an open neighbourhood of the Shilov boundary (in the boundary of the domain G). This is joint work with B. Joricke.Paper II considers smoothly bounded Reinhardt domains in 2 dimensions and complete Reinhardt domains in complex affine n-space. We prove that in many cases one may let S be the Shilov boundary. Moreover, the respective continuity properties propagate to the closure of the envelope of holomorphy of the domain.Paper III considers pseudoconvex Hartogs domains with connected vertical sections in two dimensions. We prove that in some cases S may be taken to be the Shilov boundary and give examples for when this is not enough.

A new proof and a generalization of Ramadanov's theorem

We study the Bergman kernels on an increasing sequence of domains in complex space and show that the Bergman kernels converge uniformly on compact sets. This gives a new proof of a result of Ramadanov. We then generalize the result to the case when the domains are not necessarily increasing.

Examples of unbounded homogeneous domains in complex space

We present several examples of homogeneous domains in complex space that do not have bounded realisations. The domains are equivalent to tubes over affinely homogeneous domains in real space.

Some new results on domains in complex space with non-compact automorphism group

We provide a new proof of the Wong–Rosay theorem, using the structure of the ring of holomorphic functions. As a byproduct, we provide an analogous theorem for classical bounded symmetric domains. The second main result of this article concerns a new existence theorem for holomorphic peaking functions at a hyperbolic orbit accumulation boundary point. Finally, we give a proof of a version of the Greene–Krantz conjecture using holomorphic vector fields and a strengthened Hopf lemma.

Determination of a Domain in Complex Space by its Automorphism Group

We consider results, both in one complex variable and several, which show that the algebraic or geometric structure of the automorphism group of a domain z can determine that domain. The domains considered include Ω = B, the unit ball in C n , and Ω = C n . Various illustrative examples are provided.

Biholomorphic mappings between bounded domains in C n

In this paper we prove a global boundary behavior result on biholomorphic mappings between smooth bounded domains in complex space of any dimension. As a consquence, we have that the inverse of a biholomorphic mapping between smooth bounded domains is smooth up to the boundary if the mapping is smooth up to the boundary.

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.