The geometry of complex domains is a subject with roots extending back more than a century, to the uniformization theorem of Poincaré and Koebe and the resulting proof of existence of canonical metrics for hyperbolic Riemann surfaces. In modern times, developments in several complex variables by Bergman, Hörmander, Andreotti-Vesentini, Kohn, Fefferman, and others have opened up new possibilities for the unification of complex function theory and complex geometry. In particular, geometry can be used to study biholomorphic mappings in remarkable ways. This book presents a complete picture of these developments.Beginning with the one-variable case-background information which cannot be found elsewhere in one place-the book presents a complete picture of the symmetries of domains from the point of view of holomorphic mappings. It describes all the relevant techniques, from differential geometry to Lie groups to partial differential equations to harmonic analysis. Specific concepts addressed include: - covering spaces and uniformization; Bergman geometry; automorphism groups; invariant metrics; the scaling method. All modern results are accompanied by detailed proofs, and many illustrative examples and figures appear throughout.Written by three leading experts in the field, The Geometry of Complex Domains is the first book to provide systematic treatment of recent developments in the subject of the geometry of complex domains and automorphism groups of domains. A unique and definitive work in this subject area, it will be a valuable resource for graduate students and a useful reference for researchers in the field.
Steven G. Krantz received the B.A. degree from the University of California at Santa Cruz and the Ph.D. from Princeton University. He has taught at UCLA, Princeton, Penn State, and Washington University, where he has most recently served as Chair of the Mathematics Department. Krantz has directed 18 Ph.D. Students and 9 Masters students, and is winner of the Chauvenet Prize and the Beckenbach Book Award. He edits six journals and is Editor-in-Chief of three. A prolific scholar, Krantz has published more than 55 books and more than 160 academic papers.
I. Geometric Background * Complex analysis preliminaries: Riemann Surfaces * Normal families results: B. Cartan''s Theorem (on point fixed, identity differential implies identity) * The problem of equivalence in one and several complex variables (Generalities) * The Riemann mapping theorem and the unformization theorem * Why these theorems fail in higher dimensions (e.g., why domains near the ball are typically not the ball in the circular domain case as an example) * Moduli spaces for one-dimensional domains of finite connectivity with comments on why the situation is different in several variables * The existence of local invariants of the boundary from the viewpoint of counting parameters (why Tanaka-Chern-Moser invariants exist) * Generic inequivalence of domains * Braun-Kaup-Upmeier''s Theorem on Reinhardt domains that are equivalent are linearly equivalent and thus many examples of inequivalent domains * II. The Equivalence problem from the intrinsic viewpoint * Intrinsic metrics: Why Cartan''s theorem implies that they exist in principle * Kobayashi metric basics * Caratheodory metric basics * Kahler metrics in general (notational summary--curvature of etc.) * The Bergman kernel function * The Bergman metric * Specific examples and the general results on how the metric varies with the domain (Ramadanov''s theorem) * The continuous variation with domain (Greene and Krantz)---outline of proof only * Applications of the continuous dependence of Bergman metric on the domain: closure of equivalence classes. Existence of fixed points. (include Lempert''s convexity thing here?) NO semicontinuity result--later for that * Constant curvature: Lu Qi-Keng''s theorem * III. More on the Kobayashi and Caratheodory Metrics * Invariance properties of the metrics * Applications to existence and non-existence of holomorphic mappings (Eisenman theory) * Semi-continuity of automorphism groups * IV. Automorphisms and Mappings * Generalities and the automorphism group as a Lie group * Normal families results of the general sort (some of those will be done in effect in the introduction) * The scaling method and its consequences (this might need to occupy several sections) * Bedford and Pinchuk on automorphisms of smooth boundary things in C^2 * V. Complex Manifolds * Stein manifold generalities * Invariant metrics (and when they exist) on manifolds * (this will be fairly extended--e.g. Greene and Wu conditions for the Bergman metric and their faster than quadratic negative curvature condition for Kobayashi hyperbolic) * Conditions for the automorphism group to be a Lie group * More on Riemann surfaces * Complex Manifolds with Many Automorphisms Oeljeklaus and Huckleberry theory * A Summary of classical homogeneous space results * The Mostow-Siu example