CONTENTS PRELIMINARY REMARKS........................................................................................ 5 Introduction..................................................................................................... 5 Basic definitions, examples and facts............................................................... 8 I. 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...................................... 15 II. REPRESENTATIVE COORDINATES................................................................... 16 The Bergman metric tensor......................................................................... 16 A property of representative coordinates........................................................... 19 III. AN INVARIANT DISTANCE................................................................................... 20 Biholomorphic mappings and canonical isometry................................. 20 Critical points of the invariant distance.............................................................. 22 Completeness with respect to the invariant distance..................................... 22 IV. EXTENSION THEOREM....................................................................................... 27 Semiconformal mappings........................................................................... 27 Extension theorem................................................................................................ 31 Local characterization of a biholomorphic mapping...................................... 33 V. DOMAIN DEPENDENCE...................................................................................... 36 Ramadanov theorem.................................................................................... 36 An analogue of Ramadanov theorem for decreasing sequences................ 37 A counterexample in the plane............................................................................ 39 VI. 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........................................ 54 REFERENCES.............................................................................................................. 58