Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

拟议的研究包括三个部分。(A)第一部分讨论复欧氏空间中实子流形的复切点。Gong想要研究复切集具有较大维度的实子流形。复切线的集合是实解析族。这个问题将用奇点理论中的方法来研究。希望这种方法能给复切点带来一些新的不变量。(B)第二部分讨论了由具有复切点的实曲面产生的小因子问题的最优算术条件。这个问题可以用一个全纯映射来研究，目标是找到使来自真实曲面的映射线性化的最优条件。(C)研究的第三部分是关于满足Cauchy-Riemann不等式的有限多个连续函数的公共零集的解析性。此外，Gong(与S.M.Webster)正试图获得嵌入Cauchy-Riemann流形的最小正则性以及其他相关问题。其中一个目标是简化现有的证明。所提出的研究是在多复变和局部全纯动力学领域。研究的最后一部分涉及偏微分方程解的正则性。它还涉及到纳什-莫泽快速迭代法，这是许多分析和几何问题的基础。龚方雄寻求与奇点理论有关的结果。这也可能导致拓扑不变量的一些应用。在许多研究领域中，不变量和奇点的研究对于理解物体是很重要的。当小因子或共振存在时，分类问题与哈密顿系统的研究有很强的联系，哈密顿系统在天体力学中有应用。

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.