Minimal stretch maps between hyperbolic surfaces

Details Minimal stretch maps between hyperbolic surfaces Minimal stretch maps between hyperbolic surfaces

1998-01-09

arxiv.org/abs/math/9801039v1

Mathematics

Geometric Topology

53 pages, 11 figures, version of 1986 preprint

Scientific paper

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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