The Schwarzian derivative and measured laminations on Riemann surfaces

Details The Schwarzian derivative and measured laminations on Riemann surfaces The Schwarzian derivative and measured laminations on Riemann surfaces

2005-10-18

arxiv.org/abs/math/0510365v3

Mathematics

Differential Geometry

36 pages, 5 figures; v3: important changes to account for correction to math.DG/0501194, also simplified some arguments

Scientific paper

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 $2\pi$-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.

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