Abstract
We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface. Then we introduce a new metric on the Teichmüller space by using the maximum norm on the function space. We prove that the identity map from the Teichmüller space equipped with the Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the Teichmüller metric, is continuous (but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmüller metric on the Teichmüller space. Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmüller space.
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References
Ahlfors L V. Lectures on Quasiconformal Mappings, 2nd ed. University Lecture Series, vol. 38. Providence: Amer Math Soc, 2006
Bowen R. Hausdorff dimension of quasi-circles. Publ Math Inst Hautes Études Sci, 1979, 50: 11–26
Bowen R, Series C. Markov maps assocaited with Fuchsian groups. Publ Math Inst Hautes Études Sci, 1979, 50: 153–170
Doaudy A, Earle C. Conformally natural extension of homeomorphisms of the circle. Acta Math, 1986, 157: 23–48
Earle C, McMullen C. Quasiconformal isotopies. In: Holomorphic Functions and Moduli I. MSRI Publications, vol. 10. Heidelberg: Springer-Verlag, 1988, 143–154
Gardiner F, Jiang Y, Wang Z. Holomorphic Motions and Related Topics. In: Proceedings of the Conference on Geometry of Riemann Surfaces at Anogia. London Math Society Lecture Notes Series. London: London Math Soc, 2007
Jiang Y. Geometry of geometrically finite one-dimensional maps. Comm Math Phys, 1993, 156: 639–647
Jiang Y. Smooth classification of geometrically finite one-dimensional maps. Trans Amer Math Soc, 1996, 348: 2391–2412
Jiang Y. Renormalization and Geometry in One-Dimensional and Complex Dynamics. Advanced Series in Nonlinear Dynamics, vol. 10. River Edge: World Scientific, 1996
Jiang Y. On rigidity of one-dimensional maps. Contemp Math, 1997, 211: 319–431
Jiang Y. Nanjing Lecture Notes in Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. FIM Preprint Series, ETH-Zurich, http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/JiangNJLectureIFM.pdf, 2000
Jiang Y. Differentiable rigidity and smooth conjugacy. Ann Acad Sci Fenn Math, 2005, 30: 361–383
Jiang Y. Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials. ArX-iv:0804.3104v2, 2008
Jiang Y. Differential rigidity and applications in one-dimensional dynamics. In: Dynamics, Games and Science I. Springer Proceedings in Mathematics, vol. 1. Berlin-Heidelberg: Springer, 2011, 487–502
Lehto O. Univalent Functions and Teichmüller Spaces. New York-Berlin: Springer-Verlag, 1987
McMullen C. Thermodynamics, dimension and the Weil-Petersson metric. Invent Math, 2008, 173: 365–425
Royden H. Automorphisms and isometries of Teichmüller space. In: Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings. Bucharest: Publ House of the Acad of the Socialist Republic of Romania, 1971, 273–286
Tukia P. Differentiability and rigidity of Möbius groups. Invent Math, 1985, 82: 557–578
Acknowledgements
This work was supported by the National Science Foundation of USA (Grant No. DMS-1747905), a collaboration grant from the Simons Foundation (Grant No. 523341), the Professional Staff Congress of the City University of New York Award (Grant No. PSC-CUNY 66806-00 44) and National Natural Science Foundation of China (Grant No. 11571122). This work was partially done when the author visited the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France and when he visited the Academy of Mathematics and Systems Science and the Morningside Center of Mathematics at the Chinese Academy of Sciences in Beijing, China. The author thanks these institutions for their hospitality. The author also thanks Professor Curtis McMullen for his helpful comments and suggestions for the first version of this paper and for sending to the author his recent paper [16] which led to Section 9 of this paper.
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Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday
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Jiang, Y. A function model for the Teichmüller space of a closed hyperbolic Riemann surface. Sci. China Math. 62, 2249–2270 (2019). https://doi.org/10.1007/s11425-019-9520-4
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DOI: https://doi.org/10.1007/s11425-019-9520-4