Comparisons of Metrics on Teichmller Space

For a Riemann surface X of conformally finite type （g, n）, let dT, dL and dpi （i = 1, 2） be the Teichmuller metric, the length spectrum metric and Thurston＇s pseudometrics on the Teichmutler space T（X）, respectively. The authors get a description of the Teichmiiller distance in terms of the Jenkins-Strebel differential lengths of simple closed curves. Using this result, by relatively short arguments, some comparisons between dT and dL, dpi （i = 1, 2） on Tε（X） and T（X） are obtained, respectively. These comparisons improve a corresponding result of Li a little. As applications, the authors first get an alternative proof of the topological equivalence of dT to any one of dL, dp1 and dp2 on T（X）. Second, a new proof of the completeness of the length spectrum metric from the viewpoint of Finsler geometry is given. Third, a simple proof of the following result of Liu-Papadopoulos is given： a sequence goes to infinity in T（X） with respect to dT if and only if it goes to infinity with respect to dL （as well as dpi （i = 1, 2））.