Lines of Minima and Teichmüller Geodesics

Abstract.

For two measured laminations ν⁺ and ν⁻ that fill up a hyperbolizable surface S and for $t\,\in\,(-\infty,\infty)$, let ${\mathcal{L}}_t$ 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 ${\mathcal{L}}_t$ and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface ${\mathcal{G}}_t$ on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, e ^tν⁺ and e ^tν⁻. By deriving additional information about the twists of ν⁺ and ν⁻ around the short curves, we estimate the Teichmüller distance between ${\mathcal{L}}_t$ and ${\mathcal{G}}_t$. We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of t.