Deflating hyperbolic surfaces and the shapes of optimal Lipschitz maps

Given two hyperbolic surfaces and a homotopy class of maps between them, Thurston proved that there always exists a representative minimizing the Lipschitz constant. While not unique, these minimizers are rigid along a geodesic lamination. In this paper, we investigate what happens in the complement of that lamination. To do this, we introduce deflations, certain optimal maps to trees which can be used to obstruct optimal maps between surfaces. Using a smooth version of the orthogeodesic foliation of the first author and Farre, we also construct many new families of optimal maps, showing that the obstructions coming from deflations are essentially the only ones.