Comparisons of metrics on Teichmüller space

Abstract

For a Riemann surface X of conformally finite type (g, n), let d _T, d _L and $d_{P_i } $ (i = 1, 2) be the Teichmüller metric, the length spectrum metric and Thurston’s pseudometrics on the Teichmüller space T(X), respectively. The authors get a description of the Teichmüller distance in terms of the Jenkins-Strebel differential lengths of simple closed curves. Using this result, by relatively short arguments, some comparisons between d _T and d _L, $d_{P_i } $ (i = 1, 2) on T _ɛ(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 d _T to any one of d _L, $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 d _T if and only if it goes to infinity with respect to d _L (as well as $d_{P_i } $ (i = 1, 2)).

Keywords

Length spectrum metric / Teichmüller metric / Thurston’s pseudo-metrics

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