Best Lipschitz Maps and Transverse Measures - Lecture 1

Explore a comprehensive lecture on best Lipschitz maps and transverse measures delivered by Karen Uhlenbeck from the University of Texas at Austin. Delve into Thurston's 1996 proposal for a Teichmuller space based on best Lipschitz maps between hyperbolic surfaces, and the subsequent work by Uhlenbeck and her collaborator George Daskalopoulos. Begin with an introduction to best Lipschitz extensions, followed by a discussion on infinity harmonic maps achieving the best Lipschitz constant in homotopy classes between manifolds and S1. Discover the surprising emergence of dual measures supported on geodesic laminations and their correspondence to Thurston's transverse measures. Progress to a scheme for finding maps that realize the best Lipschitz constant between hyperbolic surfaces, utilizing Schatten-von Neumann norm approximations. Examine how these approximations lead to transverse measures with Lie algebra bundle values, providing an analytic description of infinitesimal earthquakes along canonical laminations. Gain insights into the geometric realization of the theorem connecting length variation to earthquakes in Teichmuller theory during this 56-minute lecture presented at the University of Miami.