This monograph explores the problem of boundary regularity and analytic continuation of holomorphic mappings between domains in complex Euclidean spaces. Many important methods and techniques in several complex variables have been developed in connection with these questions, and the goal of this book is to introduce the reader to some of these approaches and to demonstrate how they can be used in the context of boundary properties of holomorphic maps. The authors present substantial results concerning holomorphic mappings in several complex variables with improved and often simplified proofs. Emphasis is placed on geometric methods, including the Kobayashi metric, the Scaling method, Segre varieties, and the Reflection principle.

Geometry of Holomorphic Mappings will provide a valuable resource for PhD students in complex analysis and complex geometry; it will also be of interest to researchers in these areas as a reference.

Sergey Pinchuk graduated from Moscow State University in 1971. He received his PhD (Candidate in Math) in 1974 and habilitation (Doctor of Sciences) in 1980 also from Moscow State University. He is currently a professor at Indiana University. His main work is dedicated to several complex variables. Sergey Pinchuk was an invited plenary speaker at the International Congress of Mathematicians in 1983 and 1998.

Rasul Shafikov received PhD in Mathematics in 2001 at Indiana University under the supervision of Sergey Pinchuk. He was a post-doc at Stony Brook University, New York, and a visiting scholar at the Max Planck Math Institute in Bonn, Germany. Since 2004 he is a professor at the University of Western Ontario, Canada. Rasul Shafikov is the author of over 40 papers in the area of several complex variables, complex geometry and dynamical systems.

Alexandre Sukhov was born in Ufa (Russia) in 1964. He received PhD at the Institute of Mathematics of the Russian Academy of Sciences in Ufa, 1989. From 1995 to 1999 he was maitre de conferences at the University Aix-Marseille-1. He received HDR (habilitation à diriger des recherches) from this university. From 1999 Alexandre Sukhov is a professor at the University of Lille. Alexandre Sukhov is the author of over 80 papers in complex analysis and adjacent areas, such as the non-linear analysis, PDE and symplectic geometry.

Chapter. 1. Preliminaries

Chapter. 2. Why boundary regularity?

Chapter. 3. Continuous extension of holomorphic mappings

Chapter. 4. Boundary smoothness of holomorphic mappings

Chapter. 5. Proper holomorphic mappings

Chapter. 6. Uniformization of domains with large automorphism groups

Chapter. 7. Local equivalence of real analytic hypersurfaces

Chapter. 8. Geometry of real hypersurfaces: analytic continuation

Chapter. 9. Segre varieties

Chapter. 10. Holomorphic correspondences

Chapter. 11. Extension of proper holomorphic mappings

Chapter. 12. Extension in C2

Appendix

Bibliography

Index