Graphicality, C^0 convergence, and the Calabi homomorphism

TL;DR

This paper proves that for a sequence of graphical Hamiltonian diffeomorphisms converging uniformly to the identity, their Calabi invariants tend to zero, extending previous results and exploring implications for the Calabi homomorphism in Hamiltonian homeomorphisms.

Contribution

It generalizes Oh's result to higher-dimensional symplectic manifolds using elementary methods and discusses the extension of the Calabi homomorphism to Hamiltonian homeomorphisms.

Findings

Calabi invariants of $C^0$-converging graphical Hamiltonians tend to zero.

Established a relationship between the Calabi invariant and the generalized phase function.

Extended previous two-dimensional results to more general symplectic manifolds.

Abstract

Consider a sequence of compactly supported Hamiltonian diffeomorphisms $\phi_k$ of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the $\phi_k$ $C^0$-converge to the identity then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the $\phi_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.