Singular Berezin Transforms

We give examples of pseudoconvex Reinhardt domains where the Berezin transform has integral kernel with singularities and, hence, fails to be a smoothing map. On the other hand, we show that this can never happen for a plane domain – in fact, then the Bergman kernel is always either identically zero or strictly positive everywhere on the diagonal – and also prove that, in contrast to the example by Wiegerinck from 1984, on any pseudoconvex Reinhardt domain the Bergman space can be finite-dimensional only if it reduces to the constant zero. Received: February 02, 2007. Accepted: May 28, 2007.     

Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in ℂ<Superscript>2</Superscript>

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 counterexample to the existence of a local plurisubharmonic peak function at infinity

We give an example of an unbounded pseudoconvex Reinhardt domain in , which is Kobayashi complete but admits no local plurisubharmonic peak function at infinity.

     

Deformations of strongly pseudoconvex domains

We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert’s ideas about Kobayashi extremal discs, and also has intrinsic interest.     

On Carathéodory completeness of pseudoconvex Reinhardt domains

We give a complete characterization of Carathéodory complete pseudoconvex Reinhardt domains, which extends results of Pflug, Fu and the author.

     

Estimates of Invariant Metrics on Pseudoconvex Domains Near Boundaries with Constant Levi Ranks

Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given. As a consequence, a characterization of Levi-flatness in terms of boundary behavior of the Bergman and Kobayashi metrics is obtained.     

On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains

We compute the leading and subleading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.     

Kobayashi–Royden pseudometric versus Lempert function

We give an example showing that the Kobayashi–Royden pseudometric for a pseudoconvex domain is, in general, not the derivative of the Lempert function.     

On an example concerning the Kobayashi pseudodistance

In this short note we calculate the Kobayashi pseudodistance for elementary Reinhardt domains in . They deliver us a number of examples giving a negative answer to a problem posed by S. Kobayashi.

     

Hilbert–Schmidt Hankel Operators over Complete Reinhardt Domains

Let ${\mathcal{R}}$ be an arbitrary bounded complete Reinhardt domain in ${\mathbb{C}^n}$ . We show that for ${n \geq 2}$ , if a Hankel operator with an anti-holomorphic symbol is Hilbert–Schmidt on the Bergman space ${A^2(\mathcal{R})}$ , then it must equal zero. This fact has previously been proved only for strongly pseudoconvex domains and for a certain class of pseudoconvex domains.     

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

On complete pseudoconvex Reinhardt domains in ?², we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in ?² that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite dimensional and the Hankel operator \({H_{{{\bar z}_1}{{\bar z}_2}}}\) is Hilbert-Schmidt.     

Asymptotic behavior of the Kobayashi metric in the normal direction

In this paper we construct a pseudoconvex domain in \({\mathbb{C}}^3\) where the Kobayashi metric does not blow up at a rate of one over distance to the boundary in the normal direction.     

THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.     

Effective formulas for the Carathéodory distance

For a class of Reinhardt domains we give formulas for the Carathéodory distance. As an application we discuss the product property of the Carathéodory distance when one factor domain is a Reinhardt domain of special type.     

一类拟凸域的Bergman度量与Kobayashi度量的比较定理

本文对一类拟凸域Ｅ（ｍ，ｎ，Ｋ）给出其不变Ｋａｈｌｅｒ度量下的全纯截曲率的显表达式，并构造了Ｅ（ｍ，ｎ，Ｋ）的一个不变的完备的Ｋａｈｌｅｒ度量，使得它大于或等于Ｂｅｒｇｍａｎ度量，而且其全纯截曲率的上界是一个负常数，从而得到Ｅ（ｍ，ｎ，Ｋ）的Ｂｅｒｇｍａｎ度量和Ｋｏｂａｙａｓｈｉ度量的比较定理。     

CLASSIFICATION OF A CLASS OF HYPERBOLIC REINHARDT DOMAINS

In this article, the author discusses the dimension of holomorphic automorphism groups on hyperbolic Reinhardt domains, and classifies those hyperbolic Reinhardt domains whose automorphism group has prescribed dimension n²-2 (where n is the dimension of domain).     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

Estimation on invariant distances on pseudoconvex domains of finite type in dimension two

We study the class of smooth bounded weakly pseudoconvex domains that are of finite type (in the sense of J. Kohn) and prove effective estimates on the invariant distances of Bergman and Kobayashi and also for the inner distance that is associated to the Caratheodory distance.