多复变数双全纯映射的几何函数论

多复变函数论是当代基础数学主流方向之一，而几何函数理论是其重要的组成部分。本项目运用分析、几何和代数等现代数学工具，拟研究超Cartan域和华罗庚域上凸映射的刚性及相关极值问题；综合应用多复变、泛函分析和微分方程等数学方法，拟研究多复变数有界拟凸域上Roper-Suffrdige算子和Loewner链参数表示，并应用于研究其域上双全纯映射子族的极值问题；以边界型Schwarz引理和不变度量为工具，拟研究多复变数单位球和典型域上\epsilon星形映射的增长定理、掩盖定理和偏差定理。本项目引入了新的数学工具和方法，对当前多复变数双全纯映射的若干几何和分析问题进行探索，基础性和交叉性强，不仅丰富了多复变数几何函数论的研究成果，还对于进一步揭示单复变和多复变数几何函数论之间的本质差别具有重要的理论意义。

Function theory of several complex variables is one of main directions in modern pure mathematics,geometric function theory is one of an important part in this field. Applying to analysis, geometry, algebra and other modern tools, we will study the rigidity of biholomorphic convex mapings and extremal problems in super-Cartan and Hua domains of several complex variables; With the help of several complex variables, functional Analysis and differential equation theorey, we will consider the Roper-Suffrdige operator and Loewner chains in some boundary pseudoconvex domains, and applyting to study some extremal problems for families of biholomorphic mappings；Using the boundary Schwarz lemma and invariant metric, we will study grownth theorem、covering theorem and distortion theorem of biholomorphic \epsilon starlike mappings of several complex variables in the unit ball and classical domains. Introducing new tools and methods, the project is aimed at some recent geometric and analytical problems of biholomorphic mappings of several complex variables in a basic and crossover study. The project may complete the study in geometric function theory of several complex variables, and may enable us to reveal the essential differnces between one and several complex variables.