Schwarz lemma at the boundary on the classical domain of type $\mathcal{IV}$

Jianfei Wang, Taishun Liu and Xiaomin Tang

Vol. 302 (2019), No. 1, 309–334

DOI: 10.2140/pjm.2019.302.309

Abstract

Let ${\mathcal{ℛ}}_{\mathcal{ℐ}\mathcal{V}}\left(n\right)$ be the classical domain of type $\mathcal{ℐ}\mathcal{V}$ in ${ℂ}^n$ with $n\ge 2$. The purpose of this paper is twofold. The first is to investigate the boundary points of ${\mathcal{ℛ}}_{\mathcal{ℐ}\mathcal{V}}\left(n\right)$. We give a sufficient and necessary condition such that the boundary points of ${\mathcal{ℛ}}_{\mathcal{ℐ}\mathcal{V}}\left(n\right)$ are smooth. The second is to establish the boundary Schwarz lemma on the classical domain of type $\mathcal{ℐ}\mathcal{V}$. we obtain the optimal estimates of the eigenvalues of the Fréchet derivative for holomorphic self-mappings at the smooth boundary point of ${\mathcal{ℛ}}_{\mathcal{ℐ}\mathcal{V}}\left(n\right)$.

Keywords

holomorphic mapping, Schwarz lemma at the boundary, the classical domain of type $\mathcal{IV}$

Mathematical Subject Classification 2010

Primary: 32H02

Secondary: 32H99, 30C80

Milestones

Received: 2 May 2018

Revised: 13 August 2018

Accepted: 20 January 2019

Published: 5 November 2019