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Some applications of the Bergman kernel to geometrical theory of functions. (English) Zbl 0538.32018

Complex analysis, Banach Cent. Publ. 11, 275-286 (1983).
[For the entire collection see Zbl 0529.00020.]
This is an overview of some results connected with the various applications of the Bergman kernel to different problems of geometric function theory of several complex variables. There are two main subjects: The first one is devoted to sequences of domains and their corresponding Bergman kernels, namely: (i) the question of the uniform convergence of the Bergman kernels associated to an increasing sequence of domains in \({\mathbb{C}}^ n\) (theorem 2), (ii) the same question for the case of a decreasing sequence of domains in \({\mathbb{C}}\) (theorems 3 and 4 - results of M. Skwarczynski and T. Iwinski), (iii) an application of theorem 2 to the Lu Qi-Keng conjecture due to P. Rosenthal, (iv) some modifications to the above problems adapted to the Kobayashi and Caratheodory pseudometrics due to V. Khristov (theorems 5, 6, 7 and 8). - The second subject concerns an appropriate inversion of the well-known Bremerman theorem which says that the Bergman kernel of a Cartesian product is the product of the kernels of the projection domains. A first approach is described in § 4. This part ends with a result of E. Ligocka which elucidates the same problem under a different point of view.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32F45 Invariant metrics and pseudodistances in several complex variables

Citations:

Zbl 0529.00020
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