A smooth counterexample to the Hamiltonian Seifert conjecture in R^6

TL;DR

This paper constructs a smooth counterexample in six-dimensional symplectic space showing a non-existence of periodic orbits on certain level sets, refining previous results for higher dimensions.

Contribution

It introduces a new smooth counterexample to the Hamiltonian Seifert conjecture in R^6, using a novel symplectic embedding theorem related to the horocycle flow.

Findings

Constructed a smooth proper function with a level set without periodic orbits in R^6

Refined previous constructions applicable for dimensions greater than 6

Applied a new symplectic embedding theorem to the horocycle flow

Abstract

A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of its non-singular level sets carries no periodic orbits of the Hamiltonian flow. The function can be taken to be C^0-close and isotopic to a positive-definite quadratic form so that the level set in question is isotopic to an ellipsoid. This is a refinement of previously known constructions giving such functions for 2n > 6. The proof is based on a new version of a symplectic embedding theorem applied to the horocycle flow.