Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

几何函数理论中的实子流形和全纯映射

• 批准号: 0305474
• 负责人: Xianghong Gong
• 金额: $ 11.07万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305474&HistoricalAwards=false
• 关键词: Real; Submanifolds; Holomorphic; Mappings; Geometric

Abstract: The long-term goal of this project is to study holomorphicmappings and real submanifolds arising in complex analysis andholomorphic dynamical systems. A large part of the PI's proposedresearch has its roots in Birkhoff's fixed point theorem, the KAM(Kolmogorov-Arnold-Moser) theory, and the Siegel-Bruno-Yoccoztheory. On the other hand, new research obtained by the method ofcomplex analysis will give new insights into the classicalarea-preserving maps and reversible maps, and reversible orHamiltonian systems. The PI's proposed research involves severalproblems in connection with real analytic surfaces intwo-dimensional complex space, formally linearizable reversible orarea-preserving real analytic maps, and singular Levi-flathypersurfaces in the complex projective space. The differential equations concerning the motion of N mass points, a model of the solar system,in the three-dimensional space attracting each other according to the Newton's law form a Hamiltonian system.The reversibility of a dynamical system could just be the time-symmetry that is important in practical matters. The periodic orbits of certain area-preserving mappings correspond to the periodic motion in the Hamiltonian system of the restricted three-body problem, and such studyof the existence of such periodic orbits goes back at least to work of Poincar\'e and Birkhoff on the celestial mechanics about a century ago. The PI's work aims to understand the existence of periodic orbits in such Hamiltonian or time-symmetric systems, both over the real numbers and over the complex numbers.The PI is active in undergraduate teaching, andhas also organized a graduate student seminar dealing with his research area.

摘要：这个项目的长期目标是研究复分析和全纯动力系统中的全纯映射和真实的子流形。PI的大部分研究都源于Birkhoff的不动点定理、KAM（Kolmogorov-Arnold-Moser）理论和Siegel-Bruno-Yoccoz理论。 另一方面，复分析方法的新研究将为经典的保面积映射和可逆映射以及可逆或Hamilton系统提供新的认识。PI提出的研究涉及到二维复空间中的真实的解析曲面、形式上可线性化的可逆或保面积的真实的解析映射以及复射影空间中的奇异Levi-松弛超曲面等几个问题。太阳系的模型N个质点在三维空间中按照牛顿定律相互吸引的运动微分方程组构成一个哈密顿系统，动力学系统的可逆性可能就是在实际问题中很重要的时间对称性。 某些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Poincar 'e和Birkhoff在大约世纪前的天体力学工作. PI的工作旨在了解周期轨道的存在，在这样的哈密尔顿或时间对称系统，无论是在真实的数字和复杂的numbers.The PI是活跃在本科教学，也组织了一个研究生研讨会处理他的研究领域。