Duke Mathematical Journal

The Schwarzian derivative and measured laminations on Riemann surfaces

David Dumas

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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 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. 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 CP1-structure with Fuchsian holonomy is close to a 2π-integral Jenkins-Strebel differential. We also study two compactifications of the space of CP1-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

Article information

Source
Duke Math. J. Volume 140, Number 2 (2007), 203-243.

Dates
First available in Project Euclid: 18 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1192715419

Digital Object Identifier
doi:10.1215/S0012-7094-07-14021-3

Mathematical Reviews number (MathSciNet)
MR2359819

Zentralblatt MATH identifier
1134.30035

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 57M50: Geometric structures on low-dimensional manifolds

Citation

Dumas, David. The Schwarzian derivative and measured laminations on Riemann surfaces. Duke Math. J. 140 (2007), no. 2, 203--243. doi:10.1215/S0012-7094-07-14021-3. https://projecteuclid.org/euclid.dmj/1192715419


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