Deflating hyperbolic surfaces and the shapes of optimal Lipschitz maps

TL;DR

This paper explores the structure of optimal Lipschitz maps between hyperbolic surfaces, introducing deflations to understand obstructions and constructing new families of optimal maps, thereby advancing the understanding of geometric rigidity.

Contribution

It introduces deflations as a new tool to analyze obstructions to optimal Lipschitz maps and constructs many new examples of such maps, clarifying their rigidity properties.

Findings

Deflations can obstruct optimal maps between hyperbolic surfaces.

Many new families of optimal Lipschitz maps are constructed.

Obstructions from deflations are essentially the only ones.

Abstract

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.