Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

摘要：本课题的长期目标是研究复分析和全纯动力系统中出现的全纯映射和实子流形。PI提出的大部分研究都植根于Birkhoff的不动点定理，KAM（Kolmogorov-Arnold-Moser）理论和Siegel-Bruno-Yoccoztheory。另一方面，用复分析方法获得的新研究成果将对经典的保面积映射和可逆映射以及可逆或哈密顿系统有新的认识。PI提出的研究涉及与二维复空间中的实解析曲面、形式可线性可逆的保面积实解析映射以及复射影空间中的奇异列维平面超曲面相关的几个问题。关于太阳系模型N个质点在三维空间中根据牛顿定律相互吸引的运动的微分方程形成了哈密顿系统。动力系统的可逆性可能只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与受限三体问题的哈密顿体系中的周期运动相对应，而对这种周期轨道存在性的研究至少可以追溯到一个世纪前庞加莱和伯克霍夫对天体力学的研究。PI的工作旨在了解在这样的哈密顿或时间对称系统中周期轨道的存在，包括实数和复数。他积极参与本科教学，并组织了一次研究生研讨会，讨论他的研究领域。

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.