Hodge and Teichmüller

Abstract

We consider the derivative \begin{document}$ D\pi $\end{document} of the projection \begin{document}$ \pi $\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$ \eta $\end{document} determines a relative cohomology class \begin{document}$ [\eta]_\Sigma $\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$ D\pi([\eta]_\Sigma) $\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.