In this paper we expose the impact of the fundamental discovery, made by Erik Andersén and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of a complex Euclidean space of dimensions n > 1. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present the focal points of these developments, with a view towards the future.
Dedicated to LászlóLempert in honour of his 70th birthday CONTENTS 1. Introduction 2. Stein manifolds with density properties 2.1. Density property 2.2. Volume density property 2.3. Relative density properties 2.4. Fibred density properties 2.5. Symplectic density property 3. Automorphisms with given jets 4. Fatou-Bieberbach domains 5. Twisted complex lines in C n and nonlinearizable automorphisms 6. Embedding open Riemann surfaces in C 2 7. Complex manifolds exhausted by Euclidean spaces 8. Stein manifolds with the density property and Oka manifolds 9. Complete complex submanifolds 10. An application in 3-dimensional topology 11. The recognition problem for complex Euclidean spaces References