A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbb{R}^4$

Abstract

We construct a proper $C^2$-smooth function on $\mathbb{R}^4$ 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.