本项目拟研究复单位球间的全纯逆紧映射问题及其在复几何中的应用，这类问题属于多复变函数论、CR几何和复几何的交叉前沿课题。项目组成员已经对这一领域进行了深入的研究，并取得一些有意义的研究工作。特别地，项目组成员最近在复单位球间全纯逆紧映射的分类问题和间隙现象等问题上取得了一些进展。在此基础上，本项目拟进一步研究复单位球上全纯逆紧映射的分类问题、次数最佳估计问题和复单位球上具有退化高斯映射的全纯逆紧映射问题。这些问题都是多复变函数论和CR几何中被广泛关注但尚未解决的问题。本项目希望引入一些新的思路和方法来解决这些问题，为研究相关问题提供一些新的技巧。因此，本项目拟研究的问题具有重要的理论意义和应用价值。

The project proposes to study proper holomorphic maps between complex unit balls and their applications in complex geometry, which is a topic interwined with several complex variables, CR geometry and complex geometry. The applicants already had some intensive study on this subject, and had obtained some interesting research works. Especially, the applicants recently obtained some break through on the classification problem and the gap phenomenon of proper holomorphic maps between balls. Based on these research works, the project would further study the classification problem and the optimal degree estimates problems of the proper holomorphic maps between balls, as well as te study the proper holomorphic maps between balls with degenerate Gauss maps. These problems are concerned by the mathematicians in several complex variables and CR geometry, but still unsolved up to now. The applicants would like to introduce some new ideas and methods to solve these problems, and hope that these new ideas and methods would afford some new technique for relevant problems. Hence the study of the problems in the project is quite interesting in the academic study and its applications.