Lines of Minima and Teichmüller Geodesics

Abstract

For two measured laminations ν + and ν − that fill up a hyperbolizable surface S and for t ∈ (−∞, ∞), let L t be the unique hyperbolic surface that minimizes the length function e t l(ν + ) + e −t l(ν − ) on Teichmüller space. We characterize the curves that are short in L t and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface G t on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, e t ν + and e −t ν − . By deriving additional information about the twists of ν + and ν − around the short curves, we estimate the Teichmüller distance between L t and G t . We deduce that this distance can be arbitrarily large, but that if S is a oncepunctured torus or four-times-punctured sphere, the distance is bounded independently of t.

Lines of minima are uniformly quasigeodesic

“…This paper continues the comparison between lines of minima and Teichmüller geodesics begun in [Choi et al 2006]. Let S be a hyperbolizable surface of finite type and -(S) be the Teichmüller space of S. Let ν + and ν − be two measured laminations that fill up S. The associated line of minima is the path t → ᏸ t ∈ -(S), where ᏸ t = ᏸ t (ν + , ν − ) is the unique hyperbolic surface that minimizes the length function e t l(ν + ) + e −t l(ν − ) on -(S); see [Kerckhoff 1992] and Section 2 below.…”

“…We also showed, however, that the ratio of lengths of the same short curve on the two surfaces may be arbitrarily large so the path ᏸ t may deviate arbitrarily far from Ᏻ t . It is therefore not immediately obvious how to derive Theorem A from [Choi et al 2006]. To explain our method, we first summarize the results of [Choi et al 2006] in more detail.…”

“…It is therefore not immediately obvious how to derive Theorem A from [Choi et al 2006]. To explain our method, we first summarize the results of [Choi et al 2006] in more detail. It turns out that on both ᏸ t and Ᏻ t , a curve α is short if and only if at least one of two quantities D t (α) and K t (α) is large.…”

“…We say that a curve is extremely short if it is less than some prescribed 0 > 0 depending only on the topology of S; see Section 2. The essential results in [Choi et al 2006] were the following estimates (see Section 2 for notation).…”

“…In addition to the above results from [Choi et al 2006], there are two main ingredients in the proof of Theorem A. The first is a detailed comparison of the rates of change of K t (α) and D t (α) with t. Some simple estimates are made in Lemmas 3.1 and 3.2, with more elaborate consequences drawn in Lemma 5.1 and especially Lemma 5.2.…”