Tag: 2

I'm trying to understand a more basic version of Gromov's theorem on symplectomorphisms with compact support $ D ^ 2 times D ^ 2 $ to be contractible.

Consider the dimensional disk $ D ^ 2 subset mathbb {R} ^ 2 $. Let $ omega_0 $ to be the standard symplectic form on $ mathbb {R} ^ 2 $. Let $ omega $ to be a symplectic form (not necessarily linear) on $ D ^ 2 $ such as $ omega = omega_0 $ in some neighborhoods $ U $ of $ partial D ^ 2 $. Let $ Symp _ { partial D ^ 2} (D ^ 2, omega) $ denotes the space of symplectomorphisms of $ (D ^ 2, omega) $ which equals the identity on $ U $.

Is it true that $ Symp _ { partial D ^ 2} (D ^ 2, omega) $ is contractible?

My attempt so far has been to find that by compactifiant $ D ^ 2 $ at $ S ^ 2 $such a symplectomorphism would define a diffeomorphism preserving the volume of $ S ^ 2 $ who is the identity close "$ infty $How to show that this space of diffeomorphisms is contractible? A variant of Smale's argument about Diff ($ S ^ 2 $) retract to $ SO (3) $ work here?