Transverse Measures and Best Lipschitz and Least Gradient Maps

TL;DR

This paper explores the duality between best Lipschitz (infinity harmonic) maps and least gradient maps from surfaces to the circle, linking geometric structures like geodesic laminations and transverse measures, advancing understanding of hyperbolic surface mappings.

Contribution

It establishes a duality framework connecting infinity harmonic maps and least gradient maps, with implications for Thurston's work on hyperbolic surfaces and Teichmüller space.

Findings

Infinity harmonic maps define geodesic laminations.

Least gradient maps define transverse measures.

Duality provides an analytic approach to Thurston's theories.

Abstract

We exhibit the duality between best Lipschitz (infinity harmonic) maps and least gradient maps in the case of maps from surfaces to the circle. We show that given a homotopy class of a map from a surface to the circle the infinity harmonic map defines a geodesic lamination on the surface and the dual least gradient map defines a transverse measure on the lamination. This is the initial step towards an analytic approach to Thurston's work on best Lipschitz maps between hyperbolic surfaces and Thurston's asymmetric metric on Teichmueller space.