Hodge and Teichm\"uller

Jeremy Kahn; Alex Wright

doi:10.3934/jmd.2022007

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arXiv · 2106.04665

Hodge and Teichm\"uller

Jeremy Kahn·Alex Wright

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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.

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BibTeX RIS

Jeremy Kahn, Alex Wright. 2021-06-08. Hodge and Teichm\"uller. https://doi.org/10.3934/jmd.2022007

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