Lines of minima are uniformly quasigeodesic

Abstract

We continue the comparison between lines of minima and Teichmüller geodesics begun in our previous work. For two measured laminations ν + and ν − that fill up a hyperbolizable surface S and for t ∈ (−∞, ∞), let ᏸ t be the unique hyperbolic surface that minimizes the length function e t l(ν + ) + e −t l(ν − ) on Teichmüller space. We prove that the path t → ᏸ t is a Teichmüller quasigeodesic.

A projection from filling currents to Teichmüller space

“…A common theme in Teichmüller theory is to try to minimise the length of a (sufficiently complicated) object on S over all of T (S), and to look for a metric at which the length is minimized. This idea underlies Kerckhoff's proof of the Nielsen realisation problem [Ker83](where the orbit of a large enough collection of curves is minimised), and his lines of minima [Ker92], which yield quasigeodesics in T (S) with interesting properties [CRS06,CRS07](where the sum of the lengths of two laminations that together fill the surface is minimised). Minimising the length (and estimating this minimum) for non-simple closed curves on S has also recently attracted some attention both in its own right [Bas13] and as a way to count mapping class group orbits of such curves with fixed self-intersection number [AGPS16,AS16].…”

Lines of minima with no end in Thurston’s boundary of Teichmüller space

“…It is known that there exists a line of minima between any two distinct points in T (X) (see [Ke]). It is also known that every line of minima is a uniformly quasi-geodesic for the Teichmüller distance d T (see [CRS1]); namely, there exist universal constants c ≥ 1 and C ≥ 0 such that the inequality 1 c…”