Transverse Measures and Best Lipschitz and Least Gradient Maps

Motivated by some work of Thurston on defining a Teichmuller theory based on best Lipschitz maps between surfaces, we study infinity-harmonic maps from a manifold to a circle. The best Lipschitz constant is taken on on a geodesic lamination. Moreover, in the surface case the dual problem leads to a least gradient section of a line bundle which defines a transverse measure on the lamination. We discuss the construction of least gradient sections from transverse measures.

If time permits, we will discuss what needs to be done to understand the surface to surface case.

 This is joint work with George Daskalopoulos.