Non-Relativity Of Cartan-Egg Domains And Complex Euclidean Spaces

The relativity between domains with some specific metrics and complex Euclidean spaces has always been a hot topic in the study of several complex variables in recent years.In this paper,we study that a Cartan-Egg domain with Bergman metrics is not related to a complex Euclidean space with canonical metrics.Cartan-Egg domains is a kind of very good bounded non-homogeneous domains.Its Bergman kernel can be constructed an explicit expression by the expansion principle.In the relation research of complex Euclidean spaces,the premise is that Bergman kernel is Nash algebraic function.However,the Bergman kernel of Cartan-Egg domains is not necessarily a Nash algebraic function.Therefore,the existing methods can not be used directly.In this paper,by analyzing the algebraic properties of Bergman kernel partial derivative function of a Cartan-Egg domain,it is obtained that a Cartan-Egg domain with Bergman metrics is not related to a complex Euclidean space with canonical metrics.This paper is divided into four chapters.The first chapter introduces the basic definition of Nash algebraic function and relativity,the concept of three complex space forms and transcendental degree;The second chapter introduces the definition and properties of Bergman kernel function and how to define Bergman metrics in bounded domain;The third chapter reviews the origin and definition of Cartan-Egg domains,and reviews the process of obtaining the explicit expression of Bergman kernel function of four kinds of Cartan-Egg domains by expansion principle;The fourth chapter is the main chapter of this paper,which gives the specific proof process that a Cartan-Egg domain is not related to a complex Euclidean space.