Geometric and Functional Analysis

, Volume 18, Issue 3, pp 698–754

Lines of Minima and Teichmüller Geodesics

Article

DOI: 10.1007/s00039-008-0675-6

Cite this article as:
Choi, YE., Rafi, K. & Series, C. GAFA Geom. funct. anal. (2008) 18: 698. doi:10.1007/s00039-008-0675-6

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+) + e-tl) 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, etν+ and etν. 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 

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Education Program for Gifted Youth, Ventura HallStanford UniversityStanfordUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK