Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\rightarrow\mathcal{T}$ be any diffeomorphism which preserves $\omega$.
Q) Does there exist a diffeomorphism $\phi:S\rightarrow S$ such that the induces map to $\mathcal{T}$ is $\Phi$?
Any suggestion or reference will be extremely helpful. Thanks in advance.
There are infinitely many compactly supported symplectomorphisms of any symplectic manifold, which would then have to be represented by diffeomorphisms of $S$ preserving all marked conformal structures, except those in the compact set. But preserving such a conformal structure is only possible for a finite set of diffeomorphisms, as the conformal structure has finite automorphism group.
No, the group of diffeomorphisms isotopic to the identity acts trivially, so what’s left over is the discrete mapping class group. Meanwhile, Wolpert proved that Fenchel Nielsen coordinates are global Darboux coordinates, so the symplectomorphism group of Teichmuller space is isomorphic to the symplectomorphism group of a standard $\mathbb{R}^n$. Therefore, surface diffeomorphisms give a tiny fraction of the full group of symplectomorphisms. If you also ask to preserve the complex structure, the answer is yes, at least for genus greater than two by a theorem of Royden.
