Deflating hyperbolic surfaces and the shapes of optimal Lipschitz maps

TL;DR

This work analyzes optimal Lipschitz maps between hyperbolic surfaces beyond Thurston’s tension lamination. It introduces deflations to trees as obstructions and develops a framework around a smooth orthogeodesic foliation to construct a wide range of boundary-tight maps, showing that obstructions from deflations essentially account for all constraints. The authors describe how the spine of a surface and its dual arc system encode the Lipschitz geometry, proving that long spine-edges correspond to thin bands and constructing maps on decomposed pieces (h-gons, rectangles, spikes) that can be glued to realize prescribed itineraries. The results illuminate the asymptotic behavior of Thurston geodesics, envelopes in Teichmüller space, and provide explicit, flexible methods for building Lipschitz maps with controlled boundary behavior, potentially impacting moduli problems and dynamics on Teichmüller space.

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.

Figures (14)

• Figure 1: Left: a quadrilateral that admits no immersed horogon. Middle: an embedded (and regular) horogon in the regular ideal $5$-gon. Right: an immersed horogon in a nonregular ideal $5$--gon, with one side not entirely contained in the polygon.
• Figure 2: Envelopes in the Teichmüller space of ideal pentagons. The unbounded, red, vertically-striped cones are the set of $P$ admitting a boundary-tight map from the tip of the cone (i.e., out-envelopes). The bounded, blue, horizontally-striped regions are those $P$ which map to the indicated points (i.e., in-envelopes). The center point is the regular pentagon; its out-envelope is the entire Teichmüller space, while its in-envelope (not pictured) is a small open neighborhood of the center.
• Figure 3: A lamination containing 3 simple closed curves and 4 spiraling leaves. Its complement is a union of two ideal 4-gons. There is no hyperbolic structure $X$ such that $X \setminus \lambda$ is a union of regular 4-gons, despite this satisfying the residue condition of Theorem \ref{['thm:imageofcut']}. If one were to force the complementary components to be regular, the rightmost curve would have to have length 0 (compare Proposition \ref{['prop:reg_glue_res']}).
• Figure 4: Endpoints of horocycles as in Lemma \ref{['lem:preserve horocycle endpts']}.
• Figure 5: Points and geodesics in the proof of Lemma \ref{['lem:thickthin_deflation']}

Theorems & Definitions (113)

• Theorem 1
• Theorem 2
• Corollary 3
• Remark 1.1
• Theorem 4
• Definition 2.1: Definition 2.9 of Gupta_wild
• Theorem 2.2
• Remark 2.3
• Theorem 2.4: Theorem 6.4 of shshI
• Theorem 3.1: Th_stretch