Comparisons of Metrics on Teichmller Space  

作　　者：Zongliang SUN Lixin LIU 

机构地区：[1]Department of Mathematics, Suzhou University,Suzhou 215006, Jiangsu,China [2]Department of Mathematics, Zhongshan University, Guangzhou 510275, China

出　　处：《Chinese Annals of Mathematics,Series B》2010年第1期71-84,共14页数学年刊（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China (No. 10871211)

摘　　要：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））.

关 键 词：Length spectrum metric Teichmuller metric Thurston＇s pseudo-metrics 

分 类 号：O189.11[理学—数学] O157.5[理学—基础数学]