Burns-Krantz rigidity in non-smooth domains

TL;DR

This paper extends Burns-Krantz rigidity results to non-smooth boundary points of complex domains like the polydisc and symmetrized bidisc, using invariance of complex geodesics to analyze boundary behavior.

Contribution

It proves a boundary Schwarz lemma for non-smooth boundary points in specific complex domains, advancing understanding of boundary rigidity phenomena.

Findings

Established boundary Schwarz lemma for non-smooth boundary points

Demonstrated invariance of complex geodesics at boundary points

Proposed approach for Burns-Krantz rigidity in bounded symmetric domains

Abstract

Motivated by recent papers \cite{For-Rong 2021} and \cite{Ng-Rong 2024} we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs is the phenomenon of invariance of complex geodesics (and their left inverses) being somehow regular at the boundary point under the mapping satisfying the property as in the Burns-Krantz rigidity theorem that lets the problem reduce to one dimensional problem. Additionally, we make a discussion on bounded symmetric domains and suggest a way to prove the Burns-Krantz rigidity type theorem in these domains that however cannot be applied for all bounded symmetric domains.