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