The automorphism group and limit set of a bounded domain I: the finite type case

TL;DR

This paper characterizes the automorphism groups and limit sets of bounded pseudoconvex domains of finite type, revealing their structure and smooth boundary properties, and applies these results to finite jet determination and Tits alternative theorems.

Contribution

It provides a detailed description of automorphism groups and limit sets for finite type domains, including their structure and boundary geometry, with new theorems on jet determination and Tits alternative.

Findings

Automorphism group has finitely many components and is a product of a compact and a Lie group.

Limit set is a smooth sphere-like submanifold.

Established new finite jet determination and Tits alternative theorems.

Abstract

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to ${ \rm Aut}(\mathbb{B}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.