The Schwarzian derivative and measured laminations on Riemann surfaces

Abstract

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal structure $X$ on a compact surface $S$. The main result is that these maps are nearly the same, differing by a multiplicative factor of -2 and an error term of lower order than the maps themselves (which we bound explicitly). As an application we show that the Schwarzian derivative of a $\CP^1$ structure with Fuchsian holonomy is close to a $2π$-integral Jenkins-Strebel differential. We also study compactifications of the space of $\CP^1$ structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmuller space, and we describe this extension.