  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 hyperbolic surfaces. We construct Lie algebra valued
dual functions which minimize a dual Schatten von-Neumann integral and limit on
a Lie algebra valued function v of bounded variation with least gradient
properties. The main result of the paper is that the support of the measure
which is a derivative of v lies on the canonical geodesic lamination
constructed by Thurston and further studied by Gueritaud-Kassel. In the sequel
paper we will use these results to investigate the dependence on the hyperbolic
structures and construct a variety of transverse measures. This should provide
information about the geometry and make contact with results of the Thurston
school.
