单与多复变量亚纯映射唯一性理论中若干问题

This project on the uniqueness theory of meromorphic mappings in one and several complex variables, is concerned with several branches of mathematical sciences, and begongs to an important part of frontier research in the field of complex analysis. The uniqueness theory of meromorphic mappings was originated from the well-known five-value theorem for meromorphic functions in the complex plane due to R. Nevanlinna in 1926, and then developed to the cases on Riemann surfaces and even high dimensional complex spaces due to E. M. Schmid (in 1971) and H. Fujimoto (in 1975), respectively. Up to now, a lot of researchers from Europe, America, Japan, Korean, India, Vietnam and China joined into this research field. Nowadays, there exist some open questions in one dimensional case which have an intention to being used of some crossing methods on several complex variables, geometry, equations and so on; and the case on high dimensions is in the developing stage, seems to be very difficult and has many challenges. The applicant and main members have obtained revelatively well work, and now intend to do further research on some of the important issues: the problem on the best number q for the uniqueness theory of meromorphic mappings from C^{m} into P^{n}(C) proposed by Fujimoto; the Bruck conjecture of uniqueness theory for entire functions of one complex variable; the uniqueness problems on meromorphic functions concerning total derivatives in several complex variables; the uniqueness problems on the difference operators in several complex variables. We believe that we will make some breakthrough and obtain some interesting results.

本项目研究单与多复变量亚纯映射唯一性理论，涉及多个数学分支，是现代复分析前沿研究中重要组成部分。亚纯映射唯一性理论起源于R.Nevanlina在1926年获得的平面上亚纯函数的五值定理，之后分别被E.M.Schmid（1971年）和H.Fujimoto（1975年）推广到黎曼曲面和高维空间。中国、欧美、日韩、印度、越南等国许多学者都曾加入该领域的研究工作。如今，一维中尚存许多公开难题且有结合多复变、几何、方程等交叉方法来解决的趋向；高维正处发展阶段，富有挑战。本项目申请人与主要成员已获得了较好的前期研究工作基础，我们现拟进一步重点研究其中一些重要问题：从C^{m}到P^{n}(C)的亚纯映射唯一性理论中Fujimoto提出"最佳q值"问题；单复变量整函数唯一性理论中Bruck猜想；多复变量亚纯函数涉及全导数的唯一性问题；多变量复差分算子的唯一性问题。有望取得一定突破，得到一些有意义的结果。

本项目共计发表了19篇学术论文，其中14篇SCI收录。主要在以下几个方面取得了重要学术成果，部分达到了在国际同行先进水平。..(1)我们在单与多复变亚纯映射唯一问题进行了深入研究。将杨乐院士处理重值问题的方法应用到多复变情形，对多复变到复射影空间亚纯映射涉及超平面或超曲面唯一性进行了深入研究，获得了该问题的目前国际上最好的结果。引入Rossi方法，对Bruck猜想超级等于1/2情形进行了研究。..(2)我们起始性地研究了多复变差分Nevanlinna理论，获得了固定或移动平面和超曲面的差分形式的第二基本定理，同时也获得了Picard型定理及唯一性成果。这是走在国际前沿的学术工作。..(3)利用我们之前取得的多复变差分Nevanlinna理论，对费马型偏差分、微分差分方程进行研究，获得了国际上该问题的首次成果，将推动差分Nevanlinna及复偏差分微分方程应用课题的研究。..(4)借鉴全导数概念，首次研究了多复变亚纯函数涉及全导数的正规族理论，推广了单复变正规族中经典理论。