Minimal stretch maps between hyperbolic surfaces

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.