A function model for the Teichmüller space of a closed hyperbolic Riemann surface
- 20 Downloads
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.
Keywordsdual symbolic space geometric model function model for the Teichmüller space maximum metric
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  which led to Section 9 of this paper.
- 6.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, 2007Google Scholar
- 9.Jiang Y. Renormalization and Geometry in One-Dimensional and Complex Dynamics. Advanced Series in Nonlinear Dynamics, vol. 10. River Edge: World Scientific, 1996Google Scholar
- 13.Jiang Y. Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials. ArX-iv:0804.3104v2, 2008Google Scholar
- 17.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, 273–286Google Scholar