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Comparison between Teichmüller and Lipschitz metrics
Author(s) -
Choi Young-Eun,
Rafi Kasra
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm052
Subject(s) - lipschitz continuity , mathematics , metric (unit) , metric differential , metric space , bounded function , equivalence of metrics , intrinsic metric , metric map , pure mathematics , injective metric space , space (punctuation) , distortion (music) , mathematical analysis , computer science , computer network , bandwidth (computing) , economics , operating system , amplifier , operations management
We study the Lipschitz metric on a Teichmüller space (defined by Thurston) and compare it with the Teichmüller metric. We show that in the thin part of the Teichmüller space the Lipschitz metric is approximated up to a bounded additive distortion by the sup‐metric on a product of lower‐dimensional spaces (similar to the Teichmüller metric as shown by Minsky). In the thick part, we show that the two metrics are equal up to a bounded additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in the Teichmüller space, with distances that approach zero in the Lipschitz metric while they approach infinity in the Teichmüller metric.