Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

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实子流形。PI提出的大部分研究都源于Birkhoff不动点定理、Kam(Kolmogorov-Arnold-Moser)理论和Siegel-Bruno-Yoccoz理论。另一方面，用复分析方法得到的新研究将为经典保面积映射和可逆映射以及可逆哈密顿系统提供新的见解。PI的研究涉及二维复空间中的实解析曲面、形式上可线性化的可逆实解析映射、保面积的实解析映射以及复射影空间中的奇异Levi-flathy超曲面等问题。根据牛顿定律，关于太阳系的N个质点在三维空间中相互吸引的运动的微分方程组构成了一个哈密顿系统。动力系统的可逆性可能就是在实际问题中重要的时间对称性。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，这种周期轨道的存在的研究至少可以追溯到大约一个世纪前Poincar‘e和Birkhoff关于天体力学的工作。PI的工作旨在了解这种哈密顿或时间对称系统中周期轨道的存在，无论是在实数上还是在复数上。PI在本科生教学中很活跃，还组织了一个研究生研讨会来处理他的研究领域。

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.