Automorphisms of Complex Euclidean Spaces

Abstract

In this chapter we study holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with big holomorphic automorphism groups. The main focus is the Andersen-Lempert theory and its generalization to complex manifolds enjoying Varolin’s density property. This subject is closely intertwined with Oka theory and furnishes important examples of Oka manifolds. Our knowledge of the class of Stein manifolds with the density property has advanced considerably since the first edition of this book was published, and we describe most of these new developments. Applications presented in the chapter include the construction of nonstraightenable embedded complex lines, of twisted proper holomorphic embeddings of Stein spaces into Euclidean spaces, of nonlinearizable periodic holomorphic automorphisms of \({\mathbb {C}}^{n}\), of non-Runge Fatou-Bieberbach domains, and of big families of pairwise distinct long \({\mathbb {C}}^{n}\)’s which do not admit any nonconstant holomorphic or plurisubharmonic functions.