The Schwarzian derivative and measured laminations on Riemann surfaces

2005/10/18 by Dumas, David
#30F60 (Primary) 30F45 #53C21 #57M50 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0510365

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) → 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 \CP¹ structure with Fuchsian holonomy is 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 Teichmuller space, and we describe this extension.

Related

• The Heights Theorem for infinite Riemann surfaces
• Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds
• Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds
• Earthquakes and graftings of hyperbolic surface laminations
• Analytic properties of Stretch maps and geodesic laminations