Mathematics

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

  • Xiaoliang CHENG ,
  • Bo WANG ,
  • Yihong HAO
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  • 1. College of Mathematics and Computer, Jilin Normal University, Siping, Jilin 136000, China
    2. School of Mathematics, Northwest University, Xi’an 710127, China

Received date: 2021-08-12

  Online 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.

Cite this article

Xiaoliang CHENG , Bo WANG , Yihong HAO . Non-relativity of Cartan-Egg domains and complex Euclidean spaces[J]. Journal of East China Normal University(Natural Science), 2023 , 2023(4) : 43 -51 . DOI: 10.3969/j.issn.1000-5641.2023.04.005

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