Biholomorphic invariants related to the Bergman functions

Maciej Skwarczyński

GDML_Books, (1980), p.

Résumé

CONTENTSPRELIMINARY REMARKS........................................................................................ 5 Introduction..................................................................................................... 5 Basic definitions, examples and facts............................................................... 8I. LU QI-KENQ DOMAINS........................................................................................... 13 Some properties of Lu Qi-keng domains.................................................. 13 An example of bounded non-Lu Qi-keng domain............................................ 14 Doubly connected Lu Qi-keng domains in the plane...................................... 15II. REPRESENTATIVE COORDINATES................................................................... 16 The Bergman metric tensor......................................................................... 16 A property of representative coordinates........................................................... 19III. AN INVARIANT DISTANCE................................................................................... 20 Biholomorphic mappings and canonical isometry................................. 20 Critical points of the invariant distance.............................................................. 22 Completeness with respect to the invariant distance..................................... 22IV. EXTENSION THEOREM....................................................................................... 27 Semiconformal mappings........................................................................... 27 Extension theorem................................................................................................ 31 Local characterization of a biholomorphic mapping...................................... 33V. DOMAIN DEPENDENCE...................................................................................... 36 Ramadanov theorem.................................................................................... 36 An analogue of Ramadanov theorem for decreasing sequences................ 37 A counterexample in the plane............................................................................ 39VI. THE IDEAL BOUNDARY....................................................................................... 40 Definition of the ideal boundary................................................................... 40 Characteristic properties....................................................................................... 49 The case of bounded circular domains.............................................................. 53 Plane domains and strictly pseudoconvex domains........................................ 54REFERENCES.............................................................................................................. 58

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Zbl 0443.32014

EUDML-ID : urn:eudml:doc:268383