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Transverse Measures and Best Lipschitz and Least Gradient Maps

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.