This book is the second edition of S. Kobayashi's influential book on the theory of invariant distances and their application to questions in in the theory of mappings of complex manifolds. It serves as a fine introduction to hyperbolic complex analysis.
Although this book provides a beautiful introduction to the topic, it is not a bedside book, nor is it really accessible to undergraduates. Rather, this is a graduate level text requiring some familiarity with several complex variables, differential geometry, topology, abstract algebra, and functional anaylysis.
The book opens by considering several interesting examples that motivate the generalization of the Poincaré metric on the unit disk. In Chapter 4, an intrinsic pseudodistance is defined. This pseudodistance is the largest amongst pseudodistances which agree with the Poincaré distance on the unit disk and for which every holomorphic mapping is distance-decreasing. A complex manifold is hyperbolic if this new pseudodistance is a distance. Studying these pseudodistances leads to a way of obtaining results on complex manifolds by a purely metric (or topological) method, giving geometric insight into function theoretic results.
Once the foundations are established, the author studies the holomorphic mappings of a complex manifold into a hyperbolic manifold and how the big Picard theorem can be generalized to the setting of higher dimensional manifolds. These theorems are used in Chapter 8 when studying the relationships between hyperbolic manifolds and minimal models.
The list of unsolved problems is updated by the inclusion of ten new problems. Among the 14 problems listed in the first edition all but four are still open. Furthermore, a postscript has been added that contains supplementary remarks on the material presented in this book. The postscript was added in order "to bridge the chasm between the first edition and the author's voluminous book on Hyperbolic Complex Spaces."
Kobayashi has updated this classic book to reflect recent advances. This text serves well as a first book on this subject and continues to be a must for mathematicians interested in complex variables and complex differential geometry.