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Science China Mathematics

, Volume 62, Issue 11, pp 2249–2270 | Cite as

A function model for the Teichmüller space of a closed hyperbolic Riemann surface

  • Yunping JiangEmail author
Articles
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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.

Keywords

dual symbolic space geometric model function model for the Teichmüller space maximum metric 

MSC(2010)

37F99 32H02 

Notes

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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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsQueens College of the City University of New YorkFlushingUSA
  2. 2.Department of MathematicsThe CUNY Graduate CenterNew YorkUSA

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