Bounds on bending in terms of the Schwartzian derivative and Teichmüller distance

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.