The Schwarz Lemma at the Boundary of the Symmetrized Bidisc

TL;DR

This paper investigates the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc, a complex domain with unique boundary properties, providing new insights into its boundary behavior and metric characteristics.

Contribution

It establishes a novel boundary Schwarz lemma for the symmetrized bidisc, significantly differing from previous results and addressing the domain's complex boundary structure.

Findings

Derived a new boundary Schwarz lemma for ${ extbf{G}}_2$

Provided insights into the boundary behavior of Carathéodory and Kobayashi metrics

Enhanced understanding of holomorphic self-maps on the symmetrized bidisc

Abstract

The symmetrized bidisc ${\textbf{G}}_{2}$ is defined by $${\textbf{G}}_{2}:=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1,\; z_1,z_2\in\mathbb{C}\}.$$ It is a bounded inhomogeneous pseudoconvex domain without $\mathcal{C}^1$ boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carath\'eodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc ${\textbf{G}}_2$, and our boundary Schwarz lemma in the paper differs greatly from the earlier related results.