The first thirty years of Andersen-Lempert theory

TL;DR

This paper reviews the 30-year impact of Andersen-Lempert theory, highlighting its role in advancing complex analysis and geometry through the study of automorphism groups of Stein and affine algebraic manifolds.

Contribution

It surveys the development of the theory since 1992, emphasizing new phenomena and results in complex analysis and geometry related to automorphism groups.

Findings

Density of shear-generated automorphisms in complex Euclidean spaces

Development of new results in Stein manifold automorphisms

Advances in affine algebraic manifold automorphism theory

Abstract

In this paper we expose the impact of the fundamental discovery, made by Erik Anders\'en and L\'aszl\'o Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of complex Euclidean spaces 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 some focal points of these developments, with a view towards the future.