Comparisons of Metrics on Teichmller Space

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 Teichmller metric,the length spectrum metric and Thurston’s pseudometrics on the Teichmller space T (X),respectively.The authors get a description of the Teichmller distance in terms of the Jenkins-Strebel di?erential 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 ε (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)).