Hodge and Teichmüller

Abstract

We consider the derivative $D\pi$ of the projection $\pi$ from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form $\eta$ determines a relative cohomology class $[\eta {]}_{\Sigma }$, which is a tangent vector to the stratum. We give an integral formula for the pairing of of $D\pi ([\eta {]}_{\Sigma })$ 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.

References in corpus (2)

• Effective mapping class group dynamics II: Geometric intersection numbers
• Effective mapping class group dynamics III: Counting filling closed curves on surfaces