# Periodic orbits in virtually contact structures

## TL;DR

This paper establishes the existence of non-constant contractible periodic solutions in certain magnetic Hamiltonian systems on hyperbolic surface products, using a novel holomorphic curve theory in non-compact contact manifolds.

## Contribution

It develops a new theory of holomorphic curves in symplectizations of non-compact contact manifolds arising from virtually contact structures, enabling the analysis of periodic solutions.

## Key findings

- Existence of periodic solutions below Mañé critical value
- Development of holomorphic curve theory in non-compact settings
- Application to magnetic Hamiltonian systems on hyperbolic surfaces

## Abstract

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Ma\~n\'e critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.