On three-dimensional Reeb flows: implied existence of periodic orbits and a dynamical characterization of solid torus.

Abstract

The goal of this project is (1) to extend certain results of the theory of pseudo-holomorphic curves in simplectization of contact 3-manifolds and (2) to study the aspects of topological Reeb flow on a contact 3-manifold. The theory of simplectization, initiated by H. Hofer in 1993, was developed in the last 15 years by H. Hofer, K. Wysocki and E. Zehnder. One goal of this theory is to understand the Hamiltonian flow restricted to certain levels of energy, called energy levels of contact type. Some recent results of the advisor, joint with U. Hryniewicz, extend theorems of existence and non-existence of disk-like global surface of section for the Reeb dynamics on S^3. This project aims to extend these results to cases in which disk-like global surface of section may or not exist. As a consequence, to obtain topological obstructions to the Hamiltonian dynamics of energy levels in star-shaped in R^4 and a dynamical characterization of the solid torus