Mathematics – Symplectic Geometry
Scientific paper
2006-12-27
J. Gokova Geometry Topoloogy 1 (2007), 1 - 32 ; electronic
Mathematics
Symplectic Geometry
29 pages, 1 figure, Proofs of Lemma 5.1 and 5.4 are included in this version. Some minor typos are corrected. Both uniqueness
Scientific paper
In this paper, we prove that if a continuous Hamiltonian flow fixes the points in an open subset $U$ of a symplectic manifold $(M,\omega)$, 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 smooth boundary $\del M$ 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 \cite{viterbo2}. We also prove the converse of the theorem which localizes a previously known global result in symplectic topology.
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