Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

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.