Article

Geometric and Functional Analysis

, Volume 18, Issue 3, pp 698-754

Lines of Minima and Teichmüller Geodesics

  • Young-Eun ChoiAffiliated withEducation Program for Gifted Youth, Ventura Hall, Stanford University Email author 
  • , Kasra RafiAffiliated withDepartment of Mathematics, University of Chicago
  • , Caroline SeriesAffiliated withMathematics Institute, University of Warwick

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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+) + e -t l) 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 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 \({\mathcal{L}}_t\) 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.

Keywords and phrases:

Teichmüller geodesics lines of minima hyperbolic metric

AMS Mathematics Subject Classification:

30F45 30F60 32G15 57M50