How do you measure the difference between two hyperbolic surfaces? In the 80s, Thurston proposed a new version of Teichmüller theory that says to look at the smallest Lipschitz constant of maps between them. He proved that maps with the best possible constant exist, and while they are not unique, minimizers are always rigid along a geodesic lamination. In this talk, I’ll describe work with Jing Tao in which we coarsely characterize what must (and what can) happen on the rest of the surface.