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I think surface diffeomorphism can be extend to a symplectomorphism but i can't describe this. Is there some reference?

If this is not turue, please tell me.

masao
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  • It's not quite clear what the question is. Do you mean, given an oriented surface $\Sigma$ and a diffeomorphism $\phi: \Sigma \to \Sigma$, is there always some symplectic form $\omega \in \Gamma(\bigwedge^2 T^\Sigma)$ such that $\phi^ \omega = \omega$? – Travis Willse Apr 18 '23 at 02:27
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Apr 18 '23 at 02:50
  • @TravisWillse I want to ask that is there some perturbation of given diffeomorphism to symplectomorphism. More precisely, for given diffeomorphism it is not volume preserving for some submanifolds, can we deform it into symplectomorphism? – masao Apr 18 '23 at 03:35
  • It's still not clear to me what you're asking. Is your setting that you have a surface $\Sigma$ equipped with a volume form/symplectic form $\omega$ and a diffeomorphism $\phi: \Sigma \to \Sigma$, and you want to know whether $\phi$ can be smoothly deformed to another diffeomorphism $\tilde\phi: \Sigma \to \Sigma$ such that $\tilde\phi^*\omega = \omega$ or, put another way, that each connected component of the diffeomorphism group $\operatorname{Diff}(\Sigma)$ contains a symplectomorphism of $\omega$? – Travis Willse Apr 18 '23 at 13:09
  • Or is the diffeomorphism a map between two different surfaces, each of which is equipped with its own volume/symplectic form? (Either way, I'm not sure why you mention submanifolds here.) – Travis Willse Apr 18 '23 at 13:10

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