Transverse measures and best Lipschitz and least gradient maps

Abstract

Motivated by work of Thurston on defining a version of Teichmüller theory based on best Lipschitz maps between surfaces, we study infinity-harmonic maps from a hyperbolic manifold to the circle. The best Lipschitz constant is taken on a geodesic lamination. Moreover, in the surface case the dual problem leads to a function of least gradient which defines a transverse measure on the lamination. We also discuss the construction of least gradient functions from transverse measures via primitives to Ruelle–Sullivan currents.