多复变全纯映照、全纯函数及相关问题的研究

英 文 摘 要.(限3000 Characters)：. . Holomorphic mappings and holomorphic functions in several complex variables are important objects of study for function throry in several complex variables, which is one of mainstreams of current international mathematical researches, and has abundant in research contents. Our study mainly focuses on geometric function theory, which has substantial results obtained for over the past 20 years. But there are still many difficult and interesting problems that are worth researching and exploring. Basing on the recent year’work and using mathematical tools such as Lie group,function throry, functional analysis and matrix theory, we intend to establish growth、covering theorems for starlike mappings, convex mappings, which are different dimensions between defining field and value field， and investigate the corresponding problems on complex manifold ； generalize the important and interesting results of univalent functions to the holomorphic functions in several complex variables; extend the analytical characterization of convexity in one complex variable to several complex variables; establish the refinement of Schwarz-Pick lemma. Basing on preliminary research of previous years, we expect to make breakthroughs or significant progress on these issues.

多复变函数论是国际数学主流方向之一，多复变全纯映照及全纯函数是其主要研究对象，有着极其丰富的研究内容。我们主要研究其中的几何函数论。二十多年来，该领域虽取得了丰硕的研究成果，但仍有许多难而有趣的问题值得人们去研究和探索。在多年研究工作的基础上，本项目拟用李群、多复变、泛函分析及矩阵论等数学工具，建立维数不同(即定义域和像域维数不相同)的星形映照、凸映照等双全纯映照子族的增长、掩盖定理,并在复流行上就相应问题展开探索性研究；将单叶函数论中的一些重要而有趣的结果推广到多复变全纯函数；将单复变凸函数的某一等价刻画条件推广到多复变数空间；建立精细的Schwarz-Pick 引理。项目组对这些问题已有较充分的前期研究基础，有望取得突破或显著进展。

项目基本上按预定计划进行，部分预定目标已圆满完成，也有部分目标已取得重要进展。比如，成功把单复变凸函数的某一等价刻画条件推广到多复变数空间，并从几何函数论的角度系统研究了该类映照族的性质；把单叶函数论中的经典结果Fekete-Szego 不等式推广到多复变数空间；建立了单复变和多复变精细的Schwarz-Pick 引理等。本项目的完成有力地促进了多复变几何函数论的发展。