Are there exotic symplectic structures on $ S^2 $?

Question

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic symplectic structures on $ S^2$? If yes, is there a complete classification of isomorphy classes of symplectic $ S^2$'s?

Answer 1

Note that for a closed orientable surface $\Sigma$, a symplectic structure is just a choice of area form $\omega$ (since any $2$-form on $\Sigma$ is automatically closed) and a symplectomorphism $(\Sigma, \omega) \longrightarrow (\Sigma', \omega')$ is just an area-preserving diffeomorphism (with respect to the area forms $\omega$ and $\omega'$).

In this case, de Rham's theorem implies that for two symplectic structures $(\Sigma, \omega)$ and $(\Sigma, \omega')$ on $\Sigma$ to be symplectomorphic (i.e. one is the pullback of another by an area-preserving diffeomorphism), it is necessary that $[\omega] = [\omega'] \in H^2(\Sigma; \Bbb R)$. For the converse, one can apply a theorem of Moser¹ which states that a volume form is completely determined up to volume-preserving diffeomorphism by its total volume. This implies that varying a volume form on a closed manifold within its cohomology class does not change its equivalence class under volume-preserving diffeomorphism, since by Stokes' theorem $$\int_M \omega + d\eta = \int_M \omega$$ which implies $\omega = f^\ast(\omega + d\eta)$ for some volume-preserving diffeomorphism $f$ by Moser's theorem.

So, from the above, we can conclude the following.

For a closed, orientable surface $\Sigma$ and each real number $a \in \Bbb R \setminus \{0\}$, there exists a unique symplectic form $\omega_a$ on $\Sigma$ corresponding to $a \in \Bbb R \setminus \{0\}$ under the isomorphism $H^2(\Sigma; \Bbb R) \cong \Bbb R$.

This means that there are no exotic symplectic structures on any closed orientable surface; up to symplectomorphism they are all scalar multiples of, for example, the area form of total area $1$.

Remark. Note that while this result holds for all closed, orientable surfaces, it can fail in higher dimensions since symplectic forms are no longer equivalent to volume forms. For example, McDuff has constructed a $6$-manifold two symplectic forms $\omega$ and $\omega'$ on $T^2 \times S^2 \times S^2$ with $[\omega] = [\omega']$, but $\omega$ and $\omega'$ are not symplectomorphic (although they are deformation equivalent).

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1. Moser, Jürgen. On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286-294.