Transverse Measures and Best Lipschitz and Least Gradient Maps

Karen Uhlenbeck, Institute for Advanced Study

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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 of an infinity harmonic map is realized 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 as well the construction of least gradient sections from transverse measures. The terms will all be defined and explained in the talk. We list a number of interesting directions for future work. This is joint work with George Daskalopoulos.