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Invariant distances and metrics in complex analysis – revisited. (English) Zbl 1085.32005
In 1993 the authors’ book “Invariant distances and metrics in Complex Analysis” (Berlin; de Gruyter) appeared in which an introduction to the research field of invariant pseudometrics and pseudodistances induced by them was given (see the review in Zbl 0789.32001). The book made the reader familiar with the methods and results and also with many open problems (for example completeness questions) in this area.
Since then some progress has been made in the study of these problems.
The main idea of the present book is to survey the results that have been obtained in the last 10 years. The first chapter is devoted to holomorphically invariant objects. In the first two sections the authors recall the concepts of holomorphically contractible families of functions and of pseudometrics as the distances and pseudometrics of Carathéodory, Kobayashi, Azukawa, Wu, and Bergman. On special Reinhardt domains explicit formulas are given for these metrics. Since Lempert’s important works on the geometry of (strictly) convex domains (1981) it was known that the Carathéodory and Kobabyashi metrics coincide. In an own section the example of the symmetrized bidisc is discussed. It is the first example of a domain, where the Carathéodory and Kobabyashi metrics coincide, but nevertheless is not convexifiable.
Then, in the following sections generalized holomorphically contractible families are defined and their imortant properties are discussed. The family of multipole pluricomplex Green functions is an important example of such a famlily.
A large variety of examples with explicit formulas for the objects under discussion is presented in the next section. Also the reader is made acquainted with the theory of analytic discs which was developed by Poletsky and Edigarian and succesfully applied also by Larusson-Sigurdsson. The first chapter concludes with a discussion of the product property of the generalized Green function and related holomorphically invariant objects.
The object of the second chapter is the question of hyperbolicity and of completeness of the Carathéodory metric on Reinhardt domains as well as Hartogs and circular domains. Also the corresponding questions for the Kobayashi metric is discussed.
In the third and last chapter of the book the Bergman kernel and metric are studied. After the elementary definitions were given together with explicit formulas on some domains in the first section, the Lu Qi-Keng problem (i.e., the question of non-vanishing of the Bergman kernel off the diagonal) is discussed. Then a section on the problem of completeness and the related problem of Bergman exhaustiveness and their relationship to Kobayashi’s criterion on completeness follows. The case of plane domains ( in particular of Zalcman type), hyperconvex domains and Reinhardt domains is treated.
The book concludes with a rich collection of open problems. Also there is a big list of references at the end of the book.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
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