数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 839-850.doi: 10.1007/s10473-024-0304-y

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THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn

Jianfei WANG1, Yanhui ZHANG2,*   

  1. 1. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China;
    2. Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China
  • 收稿日期:2022-10-10 修回日期:2023-02-21 出版日期:2024-06-25 发布日期:2024-05-21

THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn

Jianfei WANG1, Yanhui ZHANG2,*   

  1. 1. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China;
    2. Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China
  • Received:2022-10-10 Revised:2023-02-21 Online:2024-06-25 Published:2024-05-21
  • Contact: *Yanhui ZHANG,E-mail:zhangyanhui@th.btbu.edu.cn
  • About author:Jianfei WANG,E-mail:wangjf@mail.ustc.edu.cn
  • Supported by:
    Wang's research was supported by the National Natural Science Foundation of China (12071161, 11971165) and the Natural Science Foundation of Zhejiang Province (Z24A010005). Zhang's research was supported by the National Natural Science Foundation of China (11971042).

摘要: By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively.

关键词: Schwarz lemma, Carathéodory metric;, rigidity

Abstract: By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively.

Key words: Schwarz lemma, Carathéodory metric;, rigidity

中图分类号: 

  • 32H02