The Schwarzian Derivative and Measured Laminations on Riemann Surfaces

A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a CP1 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) → Q(X) for each conformal structure X on a compact surface S. The main result is that the difference between these two maps is bounded by a constant depending only on X. As an application we show that the Schwarzian derivative of a CP1 structure with Fuchsian holonomy is uniformly close to a 2π-integral Jenkins-Strebel differential. We also study compactifications of the space of CP1 structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmüller space, and we describe this extension.