Global rigidity in CR geometry: the Schoen-Webster theorem

TL;DR

This paper surveys the development of the Schoen-Webster theorem in CR geometry, which classifies certain CR manifolds with large automorphism groups, and offers a concise geometric proof for the compact case.

Contribution

It provides a comprehensive survey of the theorem's evolution and introduces a new short geometric proof for compact CR manifolds.

Findings

Classification of CR manifolds with non-proper automorphism groups

Identification of the standard sphere and Heisenberg space as unique cases

A concise geometric proof for the compact case

Abstract

Schoen-Webster theorem asserts a pseudoconvex CR manifold whose automorphism group acts non properly is either the standard sphere or the Heisenberg space. The purpose of this paper is to survey successive works around this result and then provide a short geometric proof in the compact case.