Minimal stretch maps between hyperbolic surfaces

TL;DR

This paper introduces a new theory of Lipschitz maps between hyperbolic surfaces, constructs extremal maps and geodesics, and develops coordinates for Teichmüller space using cataclysms, enhancing geometric analysis tools.

Contribution

It develops a Lipschitz comparison framework for hyperbolic surfaces, constructs extremal maps and geodesics, and introduces cataclysms as new coordinates for Teichmüller space.

Findings

Extremal Lipschitz constant equals maximum length ratio of measured laminations.

Constructed extremal Lipschitz maps and geodesics for hyperbolic surfaces.

Introduced cataclysms as generalized earthquakes for coordinate systems.

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.