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| Comparisons of Metrics on Teichm¨uller Space |
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Citation： |
Zongliang SUN,Lixin LIU.Comparisons of Metrics on Teichm¨uller Space[J].Chinese Annals of Mathematics B,2010,31(1):71~84 |
| Page view： 1746
Net amount： 1344 |
Authors： |
Zongliang SUN; Lixin LIU; |
Foundation： |
the National Natural Science Foundation of China (No. 10871211). |
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| Abstract： |
For a Riemann surface X of conformally finite type (g, n), let dT , dL and dPi
(i = 1, 2) be the Teichm¨uller metric, the length spectrum metric and Thurston’s pseudometrics
on the Teichm¨uller space T(X), respectively. The authors get a description of the
Teichm¨uller 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 T"(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)). |
Keywords： |
Length spectrum metric, Teichm¨uller metric, Thurston’s pseudo-metrics |
Classification： |
32G15, 30F60, 32H15 |
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