The Schwarzian derivative and measured laminations on Riemann surfaces

Abstract

A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation that can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be regarded as the Schwarzian derivative of a ${\mathbb{CP}}^1$-structure, to which one can naturally associate another measured geodesic lamination using grafting.

We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism $ML(S)\to Q(X)$ for each conformal structure $X$ on a compact surface $S$. We show that these maps are nearly identical, 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 ${\mathbb{CP}}^1$-structure with Fuchsian holonomy is close to a $2\pi$-integral Jenkins-Strebel differential. We also study two compactifications of the space of ${\mathbb{CP}}^1$-structures, one of which uses the Schwarzian derivative and another of which uses grafting coordinates. The natural map between these two compactifications is shown to extend to the boundary of each fiber over Teichmüller space, and we describe that extension