📝 SummarXiv

Abstract

Locally-univalent maps $f: \Delta \rightarrow \hat{\mathbb{C}}$ can be parametrized by their Schwartzian derivatives $Sf$, a quadratic differential whose norm $\|Sf\|_\infty$ measures how close $f$ is to being M\"obius. In particular, by Nehari, if $\|Sf\|_\infty < 1/2$ then $f$ is univalent and if $f$ is univalent then $\|Sf\|_\infty < 3/2$. Thurston gave another parametrization associating to $f$ a bending measured lamination $\beta_f$ which has a natural norm $\|\beta_f\|_L$. In this paper, we give an explicit bound on $\|\beta_f\|_L$ as a function of $\|Sf\|_\infty$ for $\|Sf\|_\infty < 1/2$. One application is a bound on the bending measured lamination of a quasifuchsian group in terms of the Teichmuller distance between the conformal structures on the two components of the conformal boundary.