Pseudo-Orbits of Contact Forms

摘要

This paper is centered on a variational problem lacking compactness on a submanifold of the loop space of a three dimensional compact and orientable manifold M. The objects studied are M, alpha, xi, and v, where alpha is a contact form on M, xi, its Reeb vector-field, v a non-singular vector-field in the kernel of alpha. These geometrical objects are ends of flow-lines of a kind of pseudo-gradient for the functional, which when parametrized suitably explain why the Palais-Smale condition fails. There are stable and unstable manifolds for the flow related to these critical points at infinity, as well as Morse index. They are reached through a singular perturbation technique, involving the full pendulum equation with coefficients equal to functions on the manifold M. the full study of the convergence problem is completed in this paper and the main characteristics of this variational problem are singled out, including the way they change depending on v.