Real Submanifolds and Holomorphic Mappings in Several Complex Variables

Oklahoma State University, Stillwater OK

Investigators

Abstract

Abstract Award: DMS-0072003 Principal Investigator: Xianghong Gong The long term goal of this project is to study holomorphic mappings and real submanifolds arising in the field of several complex variables. One topic of proposed research is on the existence of periodic points of reversible and symplectic holomorphic mappings near an elliptic fixed point of general type and on the integrability property of holomorphic symplectic mappings via meromorphic eigenfunctions. Another topic is on the topological and analytic structure of singular Levi-flat real hypersurfaces in connections with singular complex hypersurfaces. Other topics include the structure of non-reversible conformal mappings and their connections with the non-reversibility of real analytic Hamiltonian systems. The differential equations dealing with the motion of N masspoints (a model of the solar system) in the three-dimensional space attracting each other according to Newton's law form a Hamiltonian system. The periodic orbits of certain area-preserving mappings corresponds to the periodic motion in the Hamiltonian system of the restricted three body problem, and the study of the existence of such periodic orbits goes back at least to the work of Poincare and Birkhoff about a century ago. Holomorphic symplectic mappings are natural extensions of area-preserving ones. Such an extension might allow one to apply methods in complex variables to the study of real Hamiltonian systems. Indeed, recent work on the existence of non-reversible area-preserving mappings depends on some deep knowledge of conformal mappings studied extensively in complex analysis.