Minimal stretch maps between hyperbolic surfaces

In a 1995 preprint, William Thurston outlined a Teichmueller theory for hyperbolic surfaces based on maps between surfaces which minimize the Lipschitz constant. In this paper we continue the analytic investigation into best Lipschitz maps which we began in our previous paper. In the spirit of the construction of infinity harmonic functions, we obtain best Lipschitz maps u as limits of minimizers of Schatten-von Neumann integrals in a fixed homotopy class of maps between two ...

In the Teichm\"uller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coor...

This paper is a survey about the Thurston metric on the Teichm\"uller space. The central issue is the constructions of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of Thurston. Coarse geometry and isometry rigidity of the Thurston metric, relation between the Thurston metric and the Thurston compactification are discussed. Some recent generalizations and developments of the Thurston metric are sketched.

Given two triangles whose angles are all acute, we find a homeomorphism with the smallest Lipschitz constant between them and we give a formula for the Lipschitz constant of this map. We show that on the set of pairs of acute triangles with fixed area, the function which assigns the logarithm of the smallest Lipschitz constant of Lipschitz maps between them is a symmetric metric. We show that this metric is Finsler, we give a necessary and sufficient condition for a path in t...

We first show that an earthquake of a geometrically infinite hyperbolic surface induces an asymptotically conformal change in the hyperbolic metric if and only if the measured lamination associated with the earthquake is asymptotically trivial on the surface. Then we show that the contraction along earthquake paths is continuous in the Teichm\"uller space of any hyperbolic surface. Finally, we show that if a measured lamination vanishes while approaching infinity at the rate ...

Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is "maximally stretched" by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and c...

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the Teichm\"uller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive what we believe to be a new formula for the derivativ...

This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are...

We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston's stretch maps and prove the following: (1) On the Teichmueller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and...

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichm\"{u}ller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichm\"{u}ller disk such that the two are strongly asymptotic, in the Teichm\"{u}ller metric, around the noded Riemann surface. We establish a similar comparison with plumbing deformations that open the node.