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Invariant distances and metrics in complex analysis. (English) Zbl 0789.32001
De Gruyter Expositions in Mathematics. 9. Berlin: Walter de Gruyter. xi, 408 p. (1993).
This is a comprehensive and beautifully-written book about the study of invariant pseudodistances (nonnegative functions on pairs of points) and pseudometrics (nonnegative functions on the tangent bundle) in several complex variables. Such objects are typically defined in terms of extremal problems for holomorphic mappings, and in most cases of interest it is easy to see that they satisfy a Schwarz lemma, i.e. they are contracted by holomorphic mappings. The Carathéodory and Kobayashi pseudodistances and pseudometrics are the most studied and the most useful, but there is quite a number of others (defined via extremal problems for plurisubharmonic functions, for example) whose properties and interrelationships are discussed in great detail here.
There are some surprisingly tricky problems concerning the foundations of the theory of these objects (the coincidence or non-coincidence of various topologies, for example) which are treated with great care by the authors. Indeed the clarification of such questions is one of the author’s main contributions to this subject. Other topics considered here to which the authors’ own research has contributed substantially include the computation of various metrics and distances on the annulus, and the behaviour of these objects on product domains. (In 1976 S. Kobayashi claimed that the Carathéodory pseudodistance obeys the same product property as the Kobayashi pseudodistance. This subsequently turned out to be correct, but not before the authors succeeded in proving it and in turning the whole question into an interesting subject area.)
The other topics treated here are natural choices for a book in this area. There is a version of Lempert’s work on convex domains which has as a corollary the coincidence of the Carathéodory and Kobayashi pseudodistances on such domains. The approach is via functional analysis, and is based on the ideas of an unpublished 1984 preprint of Royden and Wong which is well-known to researchers in the area. In addition there is a detailed discussion of the differentiation of pseudodistances to give pseudometrics, the integration of pseudometrics to give pseudodistances, hyperbolicity, completeness, comparison with the Bergman metric, the asymptotic behaviour on strongly pseudoconvex domains, and applications to the study of biholomorphic mappings.
The book considers only domains in $$\mathbb{C}^ n$$ and assumes a basic knowledge of several complex variables. It will be a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. There are exercises at the end of each chapter, and unsolved problems are indicated throughout the text. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.
Reviewer: I.Graham (Toronto)

##### MSC:
 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32F45 Invariant metrics and pseudodistances in several complex variables