The automorphism group and limit set of a bounded domain II: the convex case

TL;DR

This paper characterizes the automorphism groups of convex domains with smooth boundaries, showing their structure, describing the limit set topology, and establishing a gap theorem distinguishing the unit ball from other domains.

Contribution

It provides a detailed description of automorphism groups and limit sets for convex domains with smooth boundaries, including a gap theorem and classification results.

Findings

Automorphism group has finitely many components with a specific Lie group structure.

Limit set is homeomorphic to a sphere under certain conditions.

A gap theorem distinguishes the unit ball from other convex domains based on the limit set.

Abstract

For convex domains with $C^{1,\epsilon}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non-compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the limit set is the entire boundary) or the limit set has co-dimension at least two in the boundary.