Journal of East China Normal University(Natural Science) ›› 2023, Vol. 2023 ›› Issue (4): 43-51.doi: 10.3969/j.issn.1000-5641.2023.04.005

• Mathematics • Previous Articles     Next Articles

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


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.

Key words: Cartan-Egg domains, isometric embedding, Nash function, Bergman metrics

CLC Number: