Direct Products for the Hamiltonian Density Property

TL;DR

This paper demonstrates that the Hamiltonian density property is preserved under direct products of Stein manifolds and establishes this property for specific complex spaces, with applications to Hamiltonian diffeomorphisms.

Contribution

It proves the preservation of the Hamiltonian density property under direct products and establishes this property for certain complex spaces, advancing understanding of symplectic and Hamiltonian structures.

Findings

Direct product of Stein manifolds with the Hamiltonian density property also has this property.

The Hamiltonian and symplectic density properties are established for $(\\mathbb{C}^\ast)^{2n}$ and traceless Calogero--Moser spaces.

A Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space is obtained.

Abstract

We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for $(\mathbb{C}^\ast)^{2n}$ and for the so-called traceless Calogero--Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space.