Real Submanifolds and Holomorphic Mappings in Several Complex Variables

摘要奖：DMS-0072003主要研究员：龚向红这个项目的长期目标是研究几个复变量领域中的全纯映射和实子流形。研究的一个主题是关于一般椭圆不动点附近可逆辛纯映象的周期点的存在性，以及利用亚纯特征函数研究全纯辛映象的可积性。另一个主题是关于奇异Levi平坦实超曲面与奇异复超曲面的联系的拓扑学和解析结构。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不可逆性的联系。根据牛顿定律处理Nmaspoint(太阳系的一个模型)在三维空间中相互吸引的运动的微分方程形成哈密顿系统。保面积映射的周期轨道对应于限制三体问题的哈密顿系统中的周期运动，对这种周期轨道的存在性的研究至少可以追溯到大约一个世纪前的Poincare和Birkhoff的工作。全纯辛映射是保面积映射的自然推广。这样的推广可能允许人们将复变量中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的存在性的工作依赖于一些在复分析中被广泛研究的共形映射的深入知识。

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.