The first thirty years of Andersen-Lempert theory

Abstract

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 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.

References in corpus (30)

• Flexible varieties and automorphism groups
• Criteria for the density property of complex manifolds
• Bordered Riemann surfaces in C^2
• Integer homology 3-spheres admit irreducible representations in SL(2,C)
• Every bordered Riemann surface is a complete proper curve in a ball
• On exotic algebraic structures on affine spaces
• Proper holomorphic embeddings into Stein manifolds with the density property
• Embeddings of infinitely connected planar domains into C^2
• Algebraic volume density property of affine algebraic manifolds
• The algebraic density property for affine toric varieties
• Oka properties of ball complements
• A Carleman type theorem for proper holomorphic embeddings
• Complete bounded embedded complex curves in C^2
• Elliptic characterization and localization of Oka manifolds
• Complete embedded complex curves in the ball of ${\mathbb{C}}^2$ can have any topology
• Infinite transitivity on affine varieties
• Proper holomorphic immersions into Stein manifolds with the density property
• A foliation of the ball by complete holomorphic discs
• Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds
• A new notion of Tameness
• Oka properties of complements of holomorphically convex sets
• Stein Spaces Characterized by their Endomorphisms
• A parametric jet-interpolation theorem for holomorphic automorphisms of C^n
• A long ${\mathbb{C}}^2$ without holomorphic functions
• Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space
• Complete densely embedded complex lines in ${\mathbb{C}}^2$
• Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem
• Fatou-Bieberbach domains in ${\mathbb{C}}^n\setminus {\mathbb{R}}^k$
• Runge tubes in Stein manifolds with the density property
• Parametric Jet interpolation for Stein manifolds with the Density Property

Cited by in corpus (3)

• Recent developments on Oka manifolds
• Oka domains in Euclidean spaces
• Tame sets in homogeneous spaces