• 数学 •

### Cartan-Egg域与复欧氏空间的不相关性

1. 1. 吉林师范大学 数学与计算机学院, 吉林 四平　136000
2. 西北大学 数学学院, 西安　710127
• 收稿日期:2021-08-12 出版日期:2023-07-25 发布日期:2023-07-25
• 作者简介:程晓亮, 男, 教授, 博士生导师, 研究方向为复分析与数学教育. E-mail: chengxiaoliang92@163.com
• 基金资助:
国家自然科学基金 (12026420); 吉林省教育厅“十三五”科学技术项目 (JJKH20200405KJ); 吉林省科技发展计划项目 (YDZJ202201ZYTS627)

### Non-relativity of Cartan-Egg domains and complex Euclidean spaces

Xiaoliang CHENG1(), Bo WANG1, Yihong HAO2

1. 1. College of Mathematics and Computer, Jilin Normal University, Siping, Jilin　136000, China
2. School of Mathematics, Northwest University, Xi’an　710127, China
• Received:2021-08-12 Online:2023-07-25 Published:2023-07-25

Abstract:

In recent years, the relativity between domains with specific metrics and complex Euclidean spaces has been a topic of interest in the study of complex variables. Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with their induced metrics. A Cartan-Egg domain is a type of bounded non-homogeneous domain. Its Bergman kernel function can be constructed as an explicit expression using the expansion principle. In this paper, the relativity between a Cartan-Egg domain with Bergman metrics and a complex Euclidean space with canonical metrics is explored. In relation research of complex Euclidean spaces, the working premise is that a Bergman kernel function is a Nash function. However, the Bergman kernel function of Cartan-Egg domains are not necessarily Nash functions. Therefore, existing methods cannot be used directly. By analyzing the algebraic properties of a Bergman kernel function’s partial derivative function of a Cartan-Egg domain, we show that a Cartan-Egg domain with Bergman metrics is not related to a complex Euclidean space with canonical metrics.