Real Submanifolds and Holomorphic Mappings in Several Complex Variables

多个复变量中的实子流形和全纯映射

• 批准号: 0072003
• 负责人: Xianghong Gong
• 金额: $ 7.35万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072003&HistoricalAwards=false
• 关键词: Real; Submanifolds; Holomorphic; Mappings; Several

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.

摘要奖项：DMS-0072003 项目负责人：宫向红 该项目的长期目标是研究多复变量领域中出现的全纯映射和实子流形。所提出的研究主题之一是关于一般类型的椭圆不动点附近可逆和辛全纯映射的周期点的存在性以及通过亚纯本征函数的全纯辛映射的可积性质。另一个主题是奇异列维平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不可逆性的联系。根据以下方程处理三维空间中 N 个质量点（太阳系的模型）相互吸引的运动的微分方程牛顿定律形成了哈密尔顿系统。某些保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，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。这样的扩展可能允许人们将复杂变量的方法应用于真实哈密顿系统的研究。事实上，最近关于不可逆保面积映射存在性的工作依赖于在复分析中广泛研究的共形映射的一些深入知识。