Real Submanifolds and Holomorphic Mappings in Several Complex Variables

项目负责人：龚向红。本项目的长期目标是研究多复变量领域的全纯映射和实子流形。本文研究了可逆辛纯映射在一般椭圆不动点附近的周期点的存在性和全纯辛纯映射经亚纯本征函数的可积性。另一个课题是奇异列维平面实超曲面与奇异复超曲面的拓扑结构和解析结构。其他主题包括不可逆共形映射的结构及其与实解析哈密顿系统的非可逆性的联系。根据牛顿定律处理三维空间中相互吸引的nmass点（太阳系的一种模型）运动的微分方程形成了哈密顿系统。某些保持面积映射的周期轨道对应于受限三体问题的哈密顿系统中的周期运动，而对这种周期轨道存在性的研究至少可以追溯到大约一个世纪前庞加莱和伯克霍夫的工作。全纯辛映射是保面积映射的自然扩展。这种扩展可能允许人们将复变量的方法应用到实际哈密顿系统的研究中。事实上，最近关于不可逆保面积映射存在性的研究依赖于对复分析中广泛研究的保形映射的深入了解。

AbstractAward: DMS-0072003Principal Investigator: Xianghong GongThe long term goal of this project is to study holomorphicmappings and real submanifolds arising in the field of severalcomplex variables. One topic of proposed research is on theexistence of periodic points of reversible and symplecticholomorphic mappings near an elliptic fixed point of general typeand on the integrability property of holomorphic symplecticmappings via meromorphic eigenfunctions. Another topic is on thetopological and analytic structure of singular Levi-flat realhypersurfaces in connections with singular complexhypersurfaces. Other topics include the structure ofnon-reversible conformal mappings and their connections with thenon-reversibility of real analytic Hamiltonian systems.The differential equations dealing with the motion of Nmasspoints (a model of the solar system) in the three-dimensionalspace attracting each other according to Newton's law form aHamiltonian system. The periodic orbits of certainarea-preserving mappings corresponds to the periodic motion inthe Hamiltonian system of the restricted three body problem, andthe study of the existence of such periodic orbits goes back atleast to the work of Poincare and Birkhoff about a centuryago. Holomorphic symplectic mappings are natural extensions ofarea-preserving ones. Such an extension might allow one to applymethods in complex variables to the study of real Hamiltoniansystems. Indeed, recent work on the existence of non-reversiblearea-preserving mappings depends on some deep knowledge ofconformal mappings studied extensively in complex analysis.