多复变中某些特定度量下的域与复欧氏空间的相关性一直是近年来研究的热点问题. 如果两个Kähler流形具有公共的Kähler子流形, 则称它们是相关的, 否则称为不相关的. Cartan-Egg域是一类非常好的有界非齐性域, 其Bergman核函数的显表达式可以通过膨胀原理构造得到, 研究具有Bergman度量的Cartan-Egg域与具有平坦度量的复欧氏空间的相关性是有意义的. 如果一个域的Bergman核函数是Nash函数, 容易分析在其诱导的Bergman度量下与复欧氏空间的相关性, 而Cartan-Egg域的Bergman核函数不是Nash函数. 通过分析Cartan-Egg域的Bergman核函数的偏导函数的代数性质, 得到具有Bergman度量的Cartan-Egg域与具有平坦度量的复欧氏空间是不相关的.

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.