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Cited by 19Geometric and Algebraic TopologyMathematical Dynamics and FractalsAnalytic and geometric function theory

The Schwarzian derivative and measured laminations on Riemann surfaces

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Dumas, David·2007·Duke Mathematical Journal
DOI 10.1215/s0012-7094-07-14021-3
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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

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  • The space of holomorphic quadratic differentials QD on Σ_p is homeomorphic to the space of measured foliations MF on Σ_p.

    “By the famous theorem of Hubbard and Masur [31], there is a homeomorphism between the space of holomorphic quadratic differential QD and the space of measured foliations MF on Σ p .”

    From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [31]

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Authors
Dumas, D.
Affiliations
Brown University (US)
Corresponding author
Dumas, David
ORCID
Dumas, David
Venue
Duke Mathematical Journal
Publisher
Duke University Press
Published
2007-11-01
Volume
140
Issue
2
Pages
203-243
ISSN
  • 0012-7094 (ISSN-L)
  • 1547-7398
DOI
10.1215/s0012-7094-07-14021-3
OpenAlex
W2146761118
Cited by
19
References
0
Citations by year
2024: 1 · 2023: 2 · 2022: 1 · 2021: 1 · 2020: 4 · 2017: 1 · 2016: 1 · 2014: 3
FWCI
1.95
Type
journal article
Language
en
Open access
Locations
  • Duke Mathematical Journal (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
  • Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
  • Physical Sciences › Mathematics › Geometry and Topology › Analytic and geometric function theory
Keywords
Schwarzian derivative, holomorphic quadratic differentials, hyperbolic Riemann surfaces, grafting, Teichmüller space, measured foliations

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