Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

本研究共分三个部分。(a)第一部分讨论了空间中真实的子流形的复切点 复欧氏空间龚想研究复切线集维数很大的真实的子流形。复切线集是真实的解析簇。这个问题将用奇点理论的方法来研究。希望这种方法将导致一些新的不变量的复杂的切点。(b)第二部分讨论了由具有复切点的真实的曲面所产生的小因子问题的最优算法条件。这个问题可以研究 用一个全纯映射，目标是找到线性化由真实的曲面产生的映射的最佳条件。(c)第三部分研究了满足Cauchy-Riemann不等式的多个连续函数的公共零点集的解析性。此外，龚（与S。M.韦伯斯特）正试图获得嵌入柯西-黎曼流形的最小正则性及其他相关问题。目标之一是简化现有的证明，建议的研究领域是多复变量和局部全纯动力学。最后一部分的研究涉及偏微分方程及其解的计算。它还涉及Nash-Moser快速迭代法，这是分析和几何中许多问题的基础。龚寻求结果 与奇点理论有联系这也可能导致拓扑不变量的一些应用。不变量和奇点的研究对于理解许多研究领域中的对象是重要的。当存在小因子或共振时，分类问题与天体力学中应用的哈密顿系统的研究有着密切的联系。

The proposed research has three parts. (a) The first part concerns complex tangent points of real submanifolds in complex Euclidean space. Gong wants to investigate real submanifolds whose set of complex tangents has a large dimension. The sets of complex tangents are real analytic varieties. The problem will be studied by methods in the singularity theory. Hopefully this approach will lead to some new invariants for the complex tangent points. (b) The second part concerns optimal arithmetic conditions for a small divisor problem arising from real surfaces with a complex tangent point. The problem can be studied with a holomorphic mapping and the goal is to find the optimal condition to linearize the map arising from the real surfaces. (c) The third part of proposed research is about the analyticity of the common zero set of finitely many continuous functions that satisfy a Cauchy-Riemann inequality. Also, Gong (jointly with S. M. Webster) is trying to obtain minimum regularity of embedding Cauchy-Riemann manifolds and other related problems. One of goals is to simplify the existing proofs.The proposed research is in areas of several complex variables and local holomorphic dynamics. The last part of the proposed research involves partial differential equations and the regularities of their solutions. It also involves the Nash-Moser rapid iteration method which is fundamental to many problems in analysis and geometry. Gong seeks results that have connections with the singularity theory. This might also lead to some applications of topology invariants. The study of invariants and singularities is important to understand objects in many research areas. When the small divisors or resonance are presence, the classification problems have strong connections with research in Hamiltonian systems which have applications in celestial mechanics.