Topological characterizations of Hamiltonian flows on unbounded surfaces

Hamiltonian flows with finitely many singular points on compact surfaces have been characterized in the literature, and the topological invariants of such flows have been constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems. Though various fluid phenomena are modeled as flows on the plane, it is not obvious to determine whether the flows are Hamiltonian; even the set of singular points may be totally disconnected and every orbit may be contained in a straight line parallel to the $x$-axis. In fact, there are such non-Hamiltonian flows on the plane. On the other hand, this paper topologically characterizes Hamiltonian flows on unbounded surfaces and constructs their complete invariant under a regularity condition for singular points. In addition, we show that under the finite volume assumption, Hamiltonian flows on unbounded surfaces can be embedded into those on compact surfaces.