Minimal stretch maps between hyperbolic surfaces

1998/01/09 by William P. Thurston, Thurston, William P. · 27 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Morphological variations and asymmetry #math.DG #math.GT #msc:57m50

paper · pdf · doi:10.48550/arxiv.math/9801039

53 pages, 11 figures, version of 1986 preprint

arxiv created 1998/01/09 · arxiv updated 2009/11/30

Abstract

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. Cataclysms are introduced, generalizing earthquakes by permitting more violent shearing in both directions along a fault. Cataclysms provide useful coordinates for Teichmuller space that are convenient for computing derivatives of geometric function in Teichmuller space and measured lamination space.

Cited by

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• Bounding shears of spiralling triangulations on hyperbolic surfaces
• A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries
• Convergence of some horocyclic deformations to the Gardiner-Masur\n boundary
• Thurston's asymmetric metric on Margulis spacetimes
• Flexibility and degeneracy around a theorem of Thurston
• Limit cones of multi-Fuchsian representations
• On Quantizing Teichmüller and Thurston theories
• On Fuchsian loci of PSL(n,R)-Hitchin components of a pair of pants
• Divergence et parallélisme des rayons d'étirement cylindriques
• Geodesics Currents and Counting Problems
• Thurston's asymmetric metric for the space of flat metrics
• McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential
• Quantitative marked length spectrum rigidity for surfaces
• On the density of geometrically finite Kleinian groups
• Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller
• On continuity of drifts of the mapping class group
• Distinguishing Curve Types and Designer Metrics
• Asymmetry of Outer Space
• Thurston's metric on Teichmüller space and the translation lengths of mapping classes
• Ptolemy groupoids, shear coordinates and the augmented Teichmuller space
• Three-manifolds, Foliations and Circles, I
• Elements of a metric spectral theory
• Oriented graphs on curve complex I: hyperbolic and extremal length
• Some open questions on anti-de Sitter geometry
• Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichmüller space

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