Limits of Teichmüller geodesics in the Universal Teichmüller space

Abstract

A Thurston boundary of the universal Teichmüller space T(D) is the set of projective bounded measured laminations PMLbddfalse(double-struckDfalse) of double-struckD. We prove that each Teichmüller geodesic ray in T(D) converges to a unique limit point in the Thurston boundary of T(D) in the weak∗ topology. In particular, there is an open and dense set of geodesic rays, which have unique (weak*‐)limits in the Thurston boundary. We also show that the main result is sharp by providing an example of a Teichmüller geodesic, which does not converge in the uniform weak∗ topology.

Convergence of Teichmüller deformations in the universal Teichmüller space

“…In previous works [7], [8], [9], Hakobyan and the second author of this paper showed that for an integrable holomorphic quadratic differential ϕ on D, the corresponding Teichmüller geodesic of T (D) has a unique limit point on Thurston boundary of T (D). The limit point [µ ϕ ] ∈ P M L b (D) is the projective class of the transverse measure to the geodesic straightening of the vertical foliation of ϕ multiplied by the reciprocal of the length of the vertical leaves (See Remark 1.2).…”

“…If all points on S 1 are on a finite ϕ-distance from a point in D, then as δ → 0, Γ <δ B converges to the set |ν B | of horizontal trajectories of ϕ that have one endpoint in [a, b] and the other endpoint in [c, d] (See [7]). Here, a sequence of rectifiable curves γ n converges to a curve γ if there is a uniformly Lipschitz parametrizations of all curves by the same interval so that γ n converge uniformly to γ as functions.…”

“…Notice that (unlike in the case of closed surfaces) it is not simply the horizontal transverse measure induced by ϕ. In fact, almost all vertical leaves of e −iθ ϕ have finite length (in the metric | e −iθ ϕ(ζ)dζ|) and the transverse measure is scaled by the reciprocal of the length of the vertical leaves thus giving us a new type of limit when compared to closed surfaces (See [7]). Theorem 1.1 suggests that the behavior of the Teichmüller disks in T (D) is "tame" when compared to the behavior of those in the finite dimensional Teichmüller space.…”

“…Note that ν is a measured foliation of D while µ −ϕ is a measured geodesic lamination of D.Let Γ B be the family of all Jordan curves that connect [a, b] to [c, d] inside D. Let δ > 0 be fixed. Define Γ ≥δ B to be the family of all γ ∈ Γ B such that ν(γ) ≥ δ, and define Γ <δ B to be all γ ∈ Γ B such that ν(γ) < δ (See[7]). Since f s+ti does not change the y-coordinate in the canonical parameter of ϕ, we have that curves in Γ ≥δ B are mapped onto curves in Γ ≥δ fs+ti(B) and curves in Γ <δ B are mapped onto curves in Γ <δ fs+ti(B) .…”

The Heights Theorem for infinite Riemann surfaces

“…For the case when X = ∆ this map is proved to be injective by Strebel [33]. Also for X = ∆ a modular measure map (which is closely related to the horizontal measure map) is injective on the space of projective integrable holomorphic quadratic differentials with image inside the space of projective bounded measured laminations P M L b (∆) but the map is not onto (see [13]). See [34], [30], [31] or Section 2 for the definition of bounded measured laminations.…”