Graph explorer

Hodge and Teichmüller

We consider the derivative $Dπ$ of the projection $π$ from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form $η$ determines a relative cohomology class $[η]_Σ$, which is a tangent vector to the stratum. We give an integral formula for the pairing of of $Dπ([η]_Σ)$ 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.