Hyperbolic Geometry from a Local Viewpoint

A self-contained text on hyperbolic geometry for plane domains, ideal for graduate students and academic researchers.

Linda Keen (Author), Nikola Lakic (Author)

9780521863605, Cambridge University Press

Hardback, published 8 March 2007

282 pages, 32 b/w illus. 236 exercises
23.4 x 15.6 x 1.9 cm, 0.532 kg

'Here new and interesting results are collected and presented for a target audience of graduate students and researchers, but the first half of the book is well accessible also for undergraduate students, and indeed everyone who is interested in an introduction to hyperbolic geometry.' Internationale Mathematische Nachrichten

Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.

Introduction
1. Elementary transformations
2 Hyperbolic metric in the unit disk
3. Holomorphic functions
4. Topology and uniformization
5. Discontinuous groups
6 Fuchsian groups
7. General hyperbolic metric
8. The Kobayashi metric
9. The Caratheodory pseudo metric
10. Contraction properties
11. Applications
12 Applications II
13. Applications III
14. Estimating hyperbolic densities
15. Uniformly perfect domains
16 Appendix: Elliptic functions
Bibliography.

Subject Areas: Geometry [PBM]