Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GRAPHICALITY, C0 CONVERGENCE, AND THE CALABI HOMOMORPHISM

  • Received : 2016.08.25
  • Accepted : 2017.03.24
  • Published : 2017.11.30
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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.

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References

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