On the symplectic form of the moduli space of projective structures
Let \(\mathcal P(S)\) be the moduli space of isotopy classes of projective structures associated to a \(C^{\infty}\) compact connected oriented surface \(S\) with genus \(\geq2\) and let \(e: \mathcal T(S)\to\mathcal P(S)\) be the holomorphic section of the canonical surjective submersion of \(\mathcal P(S)\) onto the Teichmüller space \(\mathcal T(S)\) associated to \(S\), given by \textit{C. T. Earle} [Proc. Am. Math. Soc. 83, 527-531 (1981; Zbl 0478.32015)]. \(e\) is regarded as an intrinsic uniformization of \(\mathcal P(S)\). The authors prove \(T^*_e\Omega_P=\pi\cdot\Omega_T\) where \(\Omega_P\) and \(\Omega_T\) are natural holomorphic symplectic forms on \(\mathcal P(S)\) and \(T^*\mathcal T(S)\), respectivly. The proof depends on theorems of \textit{C. T. McMullen} [Ann. Math. 151, No.1, 327-357 (2000; Zbl 0988.32012)].
scientific article; zbMATH DE number 5507161