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I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.

Consider the dimensional disk $D^2 \subset \mathbb{R}^2$. Let $\omega_0$ be the standard symplectic form on $\mathbb{R}^2$. Let $\omega$ be any symplectic form (not necessarily linear) on $D^2$ such that $\omega = \omega_0$ in some neighbourhood $U$ of $\partial D^2$. Let $\text{Diff}_{\partial D^2}(D^2,\omega)$ denote the space of diffeomorphisms of $D^2$ such that any $\psi \in \text{Diff}_{\partial D^2}(D^2,\omega)$ satisfies $\psi^*\omega = \omega_0$ and $\psi$ is equal to the identity on $U$.

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

My attempt so far has been to note that by compactifying $D^2$ to $S^2$, any such symplectomorphism would define a diffeomorphism of $S^2$ which is the identity near "$\infty$" and takes the induced form $\tilde\omega$ to $\tilde\omega_0$ (the standard form on $S^2$). How would one go about showing that this space of diffeomorphisms is contractible?

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    $\begingroup$ First you show that this is homotopy equivalent to $\text{Diff}_{\partial D^2}(D^2)$ by Moser's trick ($\omega$ is a volume form). Then the fact that this is contractible is a theorem of Smale (now with many proofs; the original used the Poincare Bendixson theorem). $\endgroup$
    – mme
    Jun 24 '19 at 8:22
  • $\begingroup$ I presume the map $\gamma: \text{Diff}_{\partial D^2} (D^2) \to \text{Diff}_{\partial D^2} (D^2, \omega)$ is given by sending $\psi \mapsto \psi \circ \phi$ where $\phi$ is such that $\phi^*\psi^* \omega = \omega_0$.But for each $\psi$ , Moser's trick would not give you any canonical choice of $\phi$ right? How do you ensure you're choosing the $\phi$ "consistently" for different $\psi$ ? Also if you pick an arbitrary $\phi$ given by Moser, why should the map be continuous ? $\endgroup$
    – cr1t1cal
    Jun 24 '19 at 23:20
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Mike is right. I guess that you assume that $\omega$ and $\omega_0$ have the same total area $\pi$. You have to put things in the right order. Moser's lemma assures that there is a self-diffeomorphism $\psi$ of $D^2$ (everything is understood to be the identity close to $\partial D^2$), such that $\omega_0=\psi^*(\omega)$. Consider the group $G:=Diff(D^2,\omega_0)$. Then, the map $\gamma:G\to Diff(D^2,\omega)$ given by $\phi\mapsto\psi\circ\phi$ is a homeomorphism.

A short elegant way to see Smale's theorem that $G$ is contractible (although not so easy to do properly) is to consider the space $\Omega$ of the volume forms on $D^2$, equal to $\omega_0$ close to the boundary, and of total area $\pi$. The action map $$Diff(D^2)\to\Omega:\phi\mapsto\phi^*(\omega_0)$$ is surjective (Moser), and is actually a fibration (and even a locally trivial bundle: Palais 1960, Cerf 1961). The base space $\Omega$ is contractible (since it is convex!), hence the fibre $G$ over the identity has the same homotopy type as the total space $Diff(D^2)$, known to be contractible (e.g. by playing the same game with conformal structures instead of area forms, and Riemann's uniformization theorem instead of Moser's lemma...) If I'm correct, this beautiful way of thinking is due to Earle and/or Eels, but maybe it was already Smale's.

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