[Paper Review] Minimal stretch maps between hyperbolic surfaces

William P. Thurston|ArXiv.org|Jan 9, 1998

Geometric Analysis and Curvature FlowsMathematics6 references224 citations

TL;DR

This paper establishes that the minimal Lipschitz constant between two hyperbolic surfaces equals the supremum of the ratio of lengths of simple closed geodesics, providing a geometric characterization of minimal stretch maps. It introduces an asymmetric Finsler metric on Teichmüller space, constructs extremal maps via cataclysm coordinates, and proves that the maximal stretch lamination is almost always a simple closed curve.

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.

Motivation & Objective

• To develop a geometric theory of minimal stretch maps between hyperbolic surfaces analogous to Teichmüller theory.
• To characterize the minimal Lipschitz constant between two hyperbolic structures on a finite-area surface.
• To construct an asymmetric Finsler metric on Teichmüller space using extremal Lipschitz maps.
• To introduce cataclysm coordinate systems for Teichmüller space and measured lamination spaces with differentiable structure.
• To show that the maximal stretch lamination is almost always a simple closed geodesic, enabling potential computational applications.

Proposed method

• Define the Lipschitz constant of a map between hyperbolic surfaces as the essential supremum of the pointwise stretch factor.
• Prove that the minimal Lipschitz constant equals the supremum over all simple closed geodesics of the ratio of their lengths in the source and target surfaces.
• Construct extremal stretch maps using a geometric deformation process called the cataclysm, parameterized by measured laminations.
• Introduce cataclysm coordinates on Teichmüller space via inverse parametrization of measured foliations from hyperbolic structures.
• Use differentiability of the cataclysm map and uniform bounds on its derivative to establish continuity and regularity of length functions.
• Analyze the dual geometry of the unit ball in tangent and cotangent spaces, showing that the dual unit ball has almost no flat faces, indicating conical structure.

Experimental results

Research questions

• RQ1 What is the minimal Lipschitz constant for a map between two hyperbolic surfaces with fixed topological type?
• RQ2 How can extremal Lipschitz maps be constructed geometrically, and what is their relation to geodesics in the asymmetric Finsler metric?
• RQ3 What is the structure of the set of laminations achieving the maximal stretch ratio?
• RQ4 How do the tangent and cotangent spaces of Teichmüller space behave under the Lipschitz norm, particularly regarding flat faces in the unit ball?
• RQ5 Can the theory be extended to $L^p$ norms or to three-dimensional hyperbolic manifolds, particularly in the context of quasi-Fuchsian groups?

Key findings

• The minimal Lipschitz constant between two hyperbolic surfaces equals the supremum of the ratio of lengths of simple closed geodesics in the two surfaces.
• Extremal Lipschitz maps are realized as compositions of cataclysm deformations, with the maximal stretch lamination almost always being a simple closed geodesic.
• The asymmetric Finsler metric on Teichmüller space is defined by the logarithm of the stretch factor, and its geodesics correspond to one-parameter families of extremal maps.
• The tangent space of measured lamination space admits a natural identification with the space of measured foliations, and length functions are differentiable with uniformly bounded derivatives.
• The unit ball in the tangent space has almost no flat faces, indicating that the set of directions with non-trivial flatness has measure zero in the piecewise linear metric.
• The dual unit ball in the cotangent space is mostly conical, meaning that a random direction almost surely lies on a singular point with non-smooth boundary.