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arXiv · 1608.01303

Graphicality, C^0 convergence, and the Calabi homomorphism

Michael Usher
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Abstract

Consider a sequence of compactly supported Hamiltonian diffeomorphisms $ϕ_k$ 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 $ϕ_k$ $C^0$-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 $ϕ_k$. 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.

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Michael Usher. 2016-08-03. Graphicality, C^0 convergence, and the Calabi homomorphism. https://doi.org/10.4134/bkms.b160699

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