Analytic properties of Stretch maps and geodesic laminations

TL;DR

This paper investigates the analytic properties of stretch maps and geodesic laminations on hyperbolic surfaces, introducing new methods to analyze Lipschitz maps and their dual functions within Thurston's Teichmuller theory.

Contribution

It develops a novel approach to construct and analyze best Lipschitz maps as limits of p-Schatten harmonic maps, and introduces Lie algebra valued dual functions linked to Thurston's geodesic laminations.

Findings

Support of the measure dv lies on Thurston's canonical geodesic lamination.

Established existence and regularity results for p-Schatten harmonic maps.

Linked Lie algebra valued measures to Thurston's lamination structure.

Abstract

In a 1998 preprint, Bill Thurston outlined a Teichmuller theory for hyperbolic surfaces based on maps between surfaces which minimize the Lipschitz constant (minimum stretch or best Lipschitz maps). In this paper we continue the analytic investigation which we began in our previous paper. In the spirit of the construction of infinity-harmonic functions, we produce best Lipschitz maps u as limits p goes to infinity of minimizers of p-Schatten integrals (p-Schatten harmonic maps) in a fixed homotopy class between hyperbolic surfaces. We address existence and regularity of p-Schatten harmonic maps with the latter, due to higher degeneracies, being significantly harder than for ordinary p- harmonic maps. Moreover, we construct Lie algebra valued dual functions which minimize a dual q-Schatten integral and limit as q goes to 1 to a locally defined, Lie algebra valued function v of bounded variation. One of the main results of the paper is the surprising fact that the support of the measure dv (the derivative of v) lies on the canonical geodesic lamination constructed by Thurston and further studied by Gueritaud-Kassel. In the sequel paper we will show how these Lie algebra valued measures induce a transverse measure on the canonical lamination and relate to other aspects of Thurston theory.