Timeline for Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 19, 2009 at 23:21 | comment | added | Greg Kuperberg | You're very welcome; I enjoyed the question. | |
| Dec 19, 2009 at 23:01 | vote | accept | Ilya Grigoriev | ||
| Dec 19, 2009 at 23:01 | comment | added | Ilya Grigoriev | Thank you very much, this now works very well. It's a neat idea! | |
| Dec 19, 2009 at 20:54 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
added 1376 characters in body
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| Dec 19, 2009 at 18:38 | comment | added | Ilya Grigoriev | then the composition $\phi_t \circ \psi_t^-1$ will be a path of volume-preserving diffeomorphism, but the volume preserving diffeo. at time 1 will not be $\phi_1$. Is there a way to strengthen Moser to fix this and get a deformation retraction from diffeo. to volume-preserving diffeo.? Am I missing something? | |
| Dec 19, 2009 at 18:36 | comment | added | Ilya Grigoriev | Thank you for the answer! There's one thing I don't understand however: the Moser's theorem I know does not seem strong enough to give a deformation retraction from diffeo. to volume-preserving diffeo. All it would do is, given a time dependent volume form $\omega_t \in H^2(D^2)$, it'd give use a flow $\gamma_t: D^2 \to D^2$ such that $\gamma_t^*(\omega_t)=\omega_0$ and $\gamma_0 = id$. In particular, if $\omega_1 =\omega_0$, there is no guarantee that $\gamma_1 = id$. In other words, in our situation, if we have a path of diffeo $\psi_t$, apply Moser to $\psi_t^*(vol)$ to get $\gamma_t$, | |
| Dec 19, 2009 at 2:40 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |