Greene R.E., Kim K.-T., Krantz S.G. The Geometry of Complex Domains

Birkhäuser, 2011. — 210 pp.Grand visions in mathematics can begin with simple observations. It is hardly more than a homework exercise to prove that what we nowadays call the Poincaré metric on the unit disc is invariant under the biholomorphic maps of the unit disc to itself. But this easily established fact, when combined with the (profound) uniformization theorem of Poincaré and Koebe, yields the striking conclusion that, with a small number of exceptions, every Riemann surface has a canonical complete Hermitian metric of constant Gauss curvature -1_. This result became a basic tool for the study of Riemann surfaces. From this result also grew the whole subject of canonical metrics, an area which has become central in transcendental algebraic geometry and in the topology of low-dimensional manifolds.
This book is not self-contained: on occasion we use, without apology and sometimes without proof, standard results of several complex variables and in particular of the theory of the ∂ operator. Even so, we have tried to make the book as accessible as possible to the non-specialist. Most of the arguments can be followed convincingly by simply taking the unproved background results on faith, these being usually very specific and easily stated, if not easily proven. In this sense, the book will be accessible, we hope, to anyone with a basic background in complex analysis and differential geometry. We have also separated out the more technical aspects of the differential geometry so that the complex analyst can most appreciate the shape of the arguments involving curvature by simply knowing that somehow curvature attaches differential invariants to each point that must be preserved under isometries and hence preserved under biholomorphic maps. Really detailed information on differential geometry is rather seldom needed. Geodesics, for example, hardly occur in the book at all. We have tried, in short, to make almost everything accessible to as many readers as possible without short-changing the readers with more specific expertise. Brave words, but we did try.Preliminaries
Riemann Surfaces and Covering Spaces
The Bergman Kernel and Metric
Applications of Bergman Geometry
Lie Groups Realized as Automorphism Groups
The Significance of Large Isotropy Groups
Some Other Invariant Metrics
Automorphism Groups and Classification of Reinhardt Domains
The Scaling Method, I
The Scaling Method, II