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Thurston's weak metric on the Teichmüller space of the torus
Author(s):
Abdelhadi
Belkhirat;
Athanase
Papadopoulos;
Marc
Troyanov
Journal:
Trans. Amer. Math. Soc.
357
(2005),
3311-3324.
MSC (2000):
Primary 30F60, 32G15
Posted:
February 4, 2005
MathSciNet review:
2135749
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Abstract:
We define and study a natural weak metric on the Teichmüller space of the torus. A similar metric has been defined by W. Thurston on the Teichmüller space of higher genus surfaces and our definition is motivated by Thurston's definition. However, we shall see that in the case of the torus, this metric has a different behaviour than on higher genus surfaces.
References:
-
- 1.
- Busemann, H., Local metric geometry, Trans. Amer. Math. Soc. 56 (1944) 200-274. MR 0010976 (6:97f)
- 2.
- Fréchet, M. Sur quelques points de calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-74.
- 3.
- Lehto 0., Univalent functions and Teichmüller space, Springer-Verlag (1987). MR 0867407 (88f:30073)
- 4.
- Ribeiro, H. Sur les espaces à métrique faible, Portugaliae Math. 4 (1943) 21-40. MR 0009432 (5:149b)
- 5.
- Teichmüller, O. Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 22 (1939). MR 0003242 (2:187a)
- 6.
- Thurston, W.P. Minimal Stretch maps between hyperbolic surfaces, Preprint (1985) available at http://arxiv.org/abs/math.GT/9801039.
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Additional Information:
Abdelhadi
Belkhirat
Affiliation:
Département de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Address at time of publication:
King Fahd University of Petroleum and Minerals, Hail Community College, P.O. Box 2440, Hail, Saudi Arabia
Email:
belkhirat@math.u-strasbg.fr, belkhirat@hcc.kfupm.edu.sa
Athanase
Papadopoulos
Affiliation:
Département de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email:
papadopoulos@math.u-strasbg.fr
Marc
Troyanov
Affiliation:
Section de Mathématiques, École Polytechnique Féderale de Lausanne, 1015 Lausanne, Switzerland
Email:
marc.troyanov@epfl.ch
DOI:
10.1090/S0002-9947-05-03735-9
PII:
S 0002-9947(05)03735-9
Keywords:
Teichm\"uller space,
Teichm\"uller metric,
Thurston metric,
nonsymmetric metric,
weak metric,
hyperbolic plane,
quasi-isometry
Received by editor(s):
February 3, 2004
Posted:
February 4, 2005
Copyright of article:
Copyright
2005,
American Mathematical Society
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