This highly original book examines a rich tapestry of themes and concepts, including complex geometry, Finsler metrics, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.

This work examines a rich tapestry of themes and concepts and provides a comprehensive treatment of an important area of mathematics, while simultaneously covering a broader area of the geometry of domains in complex space. At once authoritative and accessible, this text touches upon many important parts of modern mathematics: complex geometry, equivalent embeddings, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.

The Geometry of Complex Domains can serve as a “coming of age” book for a graduate student who has completed at least one semester or more of complex analysis, and will be most welcomed by analysts and geometers engaged in current research.

Preface.- 1 Preliminaries.- 2 Riemann Surfaces and Covering Spaces.- 3 The Bergman Kernel and Metric.- 4 Applications of Bergman Geometry.- 5 Lie Groups Realized as Automorphism Groups.- 6 The Significance of Large Isotropy Groups.- 7 Some Other Invariant Metrics.- 8 Automorphism Groups and Classification of Reinhardt Domains.- 9 The Scaling Method, I.- 10 The Scaling Method, II.- 11 Afterword.- Bibliography.- Index.

This highly original work, written by the creators of the multivariable theory of automorphisms, is a rich tapestry of themes and concepts, and a comprehensive treatment of an important area of mathematics. From Poincaré's work on biholomorphic inequivalence in 1906, it became clear that the structures of the automorphism groups of domains in multi-dimensional complex space are more complex, and more interesting, than those in the complex plane. The authors build on this theme and trace the evolution of the classical theory to the modern theory, which is today a cornerstone of geometric analysis.

The text begins with an introductory chapter on the concept of an automorphism group in which the theory in one complex variable is presented, emphasizing the classical ideas of Schwarz, Jobe, and others. Also examined is the theory of planar domains of multiple but finite connectivity, principally develped by Heins in the 1940s and 1950s. The authors treatment progresses to the theory in several complex variables with the so-called "classical domains" of E. Cartan, the Siegel domains of type I, II, and III, and the more modern theory of automorphism groups of smoothly bounded domains.