Minimal stretch maps between hyperbolic surfaces

William P. Thurston

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Minimal stretch maps between hyperbolic surfaces

William P. Thurston

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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.

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BibTeX RIS

William P. Thurston. 1998-01-09. Minimal stretch maps between hyperbolic surfaces. https://arxiv.org/abs/math/9801039

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