Abstract |
In this paper, we prove that if a continuous
Hamiltonian flow fixes the points in an open subset U of a
symplectic manifold (M, Ω), then its associated Hamiltonian
is constant at each moment on U. As a corollary, we prove that
the Hamiltonian of compactly supported continuous Hamiltonian
flows is unique both on a compact M with contact-type boundary
∂M and on a non-compact manifold bounded at infinity. An
essential tool for the proof of the locality is the Lagrangian
intersection theorem for the conormals of open subsets proven by
Kasturirangan and the author, combined with
Viterbo's scheme that he introduced in the proof of uniqueness of
the Hamiltonian on a closed manifold. We also
prove the converse of the theorem which localizes a previously
known global result in symplectic topology.
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