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 etl(ν+) + e-tl(ν−) on Teichmuller 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 Teichmuller geodesic whose horizontal and vertical foliations are respectively, etν+ and etν−. By deriving additional information about the twists of ν+ and ν− around the short curves, we estimate the Teichmuller 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.