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.



Are symplectomorphisms of Weil–Petersson symplectic form induced from surface diffeomorphisms?
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.


