不等维多元复欧式空间中的单位球，无论是从域的角度考察其上的逆紧全纯映照问题，还是从边界的角度考察其上的CR子流形问题，都有着丰富的研究内容。众多复分析和复几何领域的专家对这些问题的研究都做过贡献。一般来说，单位球间的逆紧全纯映照问题直观，但解决起来比较困难。传统的工具和手段或者有实质性障碍，或者因为计算复杂而无法实施。 项目组以全新的手段和方法研究这些问题并取得了阶段性的成果。本项目将在此基础上继续深入研究如下三方面问题：利用射影第二基本形式和CR第二基本形式研究单位球间的逆紧全纯映照问题；探讨利用Gauss映照研究有理逆紧全纯映照问题；寻找自同构群的子群，把有理逆紧全纯映照化为多项式逆紧全纯映照，进而给出多项式映照的显示表达式。 本项目对这些问题的深入研究必将对多元复分析与复几何理论的进展和完善做出贡献。

We study the unit balls in different dimensional complex Euclidean space in several complex variables.We may consider proper holomorphic maps between the unit balls from the perspective of domain. we also may investigate the problems on the unit sphere by the view of CR submanifolds. There are abundant research contents in two fields. Many experts on sevaral complex variables and differential geometry probe into the problems.In general, it is easy relatively to understand the problems of proper holomorphic maps between the unit balls, but it is difficult to solve them.It seems very visual,however,there are some difficuties from two aspects.One is that there are substantial obstacles by the traditional tools and means.The other is that the computation is complexity and can not be operated. By using the new method, we have obtained some achievements for the study of the problems.In the project, we will study the problems in three view as follows .Firstly, We will study proper holomorphic maps between the unit balls by applying the projective second fundamental form and CR second fundamental form.Secondly,we give the relationship between Gauss mapping and proper rational holomorphic maps between the unit balls.Lastly, we obtain the way from proper rational holomorphic maps into proper polynomial holomorphic maps by finding the appropriate subgroup of the automorphism group，and write the explicit formulas of the polynomial. We will carry on the new attempt for the development and perfection of several complex variables by the deeply study on the above problems.