Real Submanifolds and Holomorphic Mappings in Several Complex Variables

AbstractAward：DMS-0072003项目负责人：龚向红本项目的长期目标是研究多复变域中的全纯映射和真实的子流形。其中一个研究主题是可逆和辛全纯映射在一般类型椭圆不动点附近周期点的存在性以及全纯辛映射通过亚纯特征函数的可积性。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拓扑和解析结构。其他主题包括不可逆共形映射的结构及其与真实的解析Hamilton系统的不可逆性的联系。处理三维空间中N个质点（太阳系模型）根据牛顿定律相互吸引的运动的微分方程形成Hamilton系统。保面积映射的周期轨道对应于限制性三体问题的Hamilton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Poincare和Birkhoff在一个世纪以前的工作。全纯辛映射是保面积辛映射的自然推广。这样一个扩展可能允许一个applymethods在复变量的研究真实的哈密顿系统。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复分析中广泛研究的共形映射的一些深入知识。

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.