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Angle geometry in the universal Teichmüller space


Authors: Jinhua Fan and Yunping Jiang
Journal: Proc. Amer. Math. Soc. 143 (2015), 1651-1659
MSC (2010): Primary 30C75, 30F60
DOI: https://doi.org/10.1090/S0002-9939-2014-12352-5
Published electronically: November 14, 2014
MathSciNet review: 3314077
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Abstract: In this paper we investigate angle geometry in the universal Teichmüller space. We construct three examples of triangles bounded by geodesic segments such that the first example has the sum of the three inner angles less than $ \pi $, the second example has the sum of the three angles equal to $ \pi $, and the third example has the sum of the three angles greater than $ \pi $. Our result gives a negative answer to a problem raised by Zhong Li and Yi Qi. Moreover, our results indicate that the universal Teichmüller space presents all hyperbolic, Euclidean, and spherical phenomena in angle geometry.


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Additional Information

Jinhua Fan
Affiliation: Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China
Email: jinhuafan@hotmail.com

Yunping Jiang
Affiliation: mathematics9 Department, Queens College of CUNY, 65-30 Kissena Blvd, Flushing, New York 11367 — and — Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, New York 10016
Email: yunping.jiang@qc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12352-5
Keywords: Teichm\"uller distance, geodesic segment, inner angle
Received by editor(s): February 21, 2013
Received by editor(s) in revised form: August 30, 2013
Published electronically: November 14, 2014
Additional Notes: The first author was partially supported by a grant from the NSF of China (No. 11201228 and No. 11371194)
The second author was partially supported by the Collaboration Grant from the Simons Foundation (No. 199837), the CUNY Collaborative Incentive Research Grant (No. 1861), awards from PSC-CUNY, a grant from the NSF of China (No. 11171121), and a collaboration grant from Academy of Mathematics and Systems Science and the Morningside Center of Mathematics at the Chinese Academy of Sciences
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.