A C^2-smooth counterexample to the Hamiltonian Seifert conjecture in R^4

TL;DR

This paper constructs a smooth counterexample in four-dimensional space showing that a Hamiltonian flow can lack periodic orbits on a regular level set, challenging the Hamiltonian Seifert conjecture.

Contribution

It provides the first detailed C^2-smooth example in R^4 where the Hamiltonian flow has no periodic orbits on a regular level set.

Findings

Constructed a proper C^2-smooth function on R^4

Hamiltonian flow has no periodic orbits on at least one regular level set

Counterexample challenges the Hamiltonian Seifert conjecture in dimension four

Abstract

We give a detailed construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.