Pinchuk Sergey, Shafikov Rasul, Sukhov Alexandre. Geometry of Holomorphic Mappings

Birkhäuser, 2023. — 217 p. — (Frontiers in Mathematics). — ISBN 978-3-031-37148-6.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.Preliminaries
Why Boundary Regularity?
Continuous Extension of Holomorphic Mappings
Boundary Smoothness of Holomorphic Mappings
Proper Holomorphic Mappings
Uniformization of Domains with Large Automorphism Groups
Local Equivalence of Real Analytic Hypersurfaces
Geometry of Real Hypersurfaces: Analytic Continuation
Segre Varieties
Holomorphic Correspondences
Extension of Proper Holomorphic Mappings
Extension in C2