Comparisons of metrics on Teichmüller space

Abstract

For a Riemann surface X of conformally finite type (g, n), let dT, dL and \( d_{P_i } \) (i = 1, 2) be the Teichmuller metric, the length spectrum metric and Thurston’s pseudometrics on the Teichmuller space T(X), respectively. The authors get a description of the Teichmuller 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, \( d_{P_i } \) (i = 1, 2) on Tɛ(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, \( d_{P_1 } \) and \( d_{P_2 } \) 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 \( d_{P_i } \) (i = 1, 2)).