Hodge and Teichm\"uller

TL;DR

This paper derives an integral formula relating the derivative of a projection from strata of differentials to Teichmüller space, comparing Hodge and Teichmüller norms to enhance understanding of hyperbolicity in Teichmüller theory.

Contribution

It provides a new integral formula for the derivative of the projection from strata to Teichmüller space and compares Hodge and Teichmüller norms, clarifying their relationship.

Findings

Derived an integral formula for the derivative of the projection.

Compared Hodge and Teichmüller norms.

Implications for understanding hyperbolicity in Teichmüller theory.

Abstract

We consider the derivative $D\pi$ of the projection $\pi$ from a stratum of Abelian or quadratic differentials to Teichm\"uller 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 Teichm\"uller space (a quadratic differential). We derive from this a comparison between Hodge and Teichm\"uller 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 Teichm\"uller theory.