# On the $C^{\infty}$ closing lemma for Hamiltonian flows on symplectic $4$-manifolds

## TL;DR

This paper proves a $C^{
abla}$$ ext{ }$closing lemma for Hamiltonian flows on 4-dimensional symplectic manifolds, extending classical results and applying to geodesic flows and Hamiltonian systems with implications in geometry and topology.

## Contribution

It establishes the $C^{
abla}$ closing lemma for Hamiltonian flows on 4-manifolds, including geodesic flows on Finsler surfaces, and extends to broader Hamiltonian systems.

## Key findings

- Proves $C^{
abla}$ closing lemma for Hamiltonian flows on 4-manifolds.
- Extends results to geodesic flows on Finsler surfaces.
- Provides applications in differential geometry and contact topology.

## Abstract

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces by combining a result of Asaoka-Irie with the dual lens map technique. Then we extend our results to Hamitonian flows with certain restriction. We also list some applications of our results in differential geometry and contact topology.