Hodge and Teichmüller

<p style='text-indent:20px;'>We consider the derivative <inline-formula><tex-math id="M1">\begin{document}$ D\pi $\end{document}</tex-math></inline-formula> of projection id="M2">\begin{document}$ \pi from a stratum Abelian or quadratic differentials to Teichmüller space. A closed one-form id="M3">\begin{document}$ \eta determines relative cohomology class id="M4">\begin{document}$ [\eta]_\Sigma $\end{document}</tex-math></inline-formula>, which is tangent vector stratum. We give an integral formula for pairing id="M5">\begin{document}$ D\pi([\eta]_\Sigma) with cotangent space (a differential). derive this comparison between Hodge and norms, has been used in work Arana-Herrera on effective dynamics mapping groups, may clarify relationship dynamical geometric hyperbolicity results theory.</p>