Angle geometry in the universal Teichmüller space
HTML articles powered by AMS MathViewer
- by Jinhua Fan and Yunping Jiang PDF
- Proc. Amer. Math. Soc. 143 (2015), 1651-1659 Request permission
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.References
- V. Božin, N. Lakic, V. Marković, and M. Mateljević, Unique extremality, J. Anal. Math. 75 (1998), 299–338. MR 1655836, DOI 10.1007/BF02788704
- Clifford J. Earle and Zhong Li, Isometrically embedded polydisks in infinite dimensional Teichmüller spaces, J. Geom. Anal. 9 (1999), no. 1, 51–71. MR 1760720, DOI 10.1007/BF02923088
- C. J. Earle, I. Kra, and S. L. Krushkal′, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927–948. MR 1214783, DOI 10.1090/S0002-9947-1994-1214783-6
- A. Fletcher and V. Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Mathematics, vol. 11, Oxford University Press, Oxford, 2007. MR 2269887
- Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906, DOI 10.1090/surv/076
- Richard S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399–406. MR 245787, DOI 10.1090/S0002-9947-1969-0245787-3
- Nikola Lakic, Strebel points, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 417–431. MR 1476999, DOI 10.1090/conm/211/02832
- Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
- Li Zhong, Nonuniqueness of geodesics in infinite-dimensional Teichmüller spaces, Complex Variables Theory Appl. 16 (1991), no. 4, 261–272. MR 1105200, DOI 10.1080/17476939108814483
- Zhong Li, Nonuniqueness of geodesics in infinite-dimensional Teichmüller spaces. II, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 355–367. MR 1234739
- Zhong Li, Non-convexity of spheres in infinite-dimensional Teichmüller spaces, Sci. China Ser. A 37 (1994), no. 8, 924–932. MR 1323829
- Li Zhong, A note on geodesics in infinite-dimensional Teichmüller spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 301–313. MR 1346814
- Li Zhong, Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1459–1465. MR 1301053, DOI 10.1090/S0002-9939-96-03164-4
- Zhong Li and Yi Qi, Fundamental inequalities of Reich-Strebel and triangles in a Teichmüller space, Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, Contemp. Math., vol. 575, Amer. Math. Soc., Providence, RI, 2012, pp. 283–297. MR 2933905, DOI 10.1090/conm/575/11398
- S. L. Krushkal, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411–418.
- Edgar Reich, Extremal quasiconformal mappings of the disk, Handbook of complex analysis: geometric function theory, Vol. 1, North-Holland, Amsterdam, 2002, pp. 75–136. MR 1966190, DOI 10.1016/S1874-5709(02)80005-1
- Edgar Reich and Kurt Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 375–391. MR 0361065
- Kurt Strebel, On the extremality and unique extremality of quasiconformal mappings of a parallel strip, Rev. Roumaine Math. Pures Appl. 32 (1987), no. 10, 923–928. MR 924137
- G. Yao, A binary infinitesimal form of Teichmüller metric, to appear.
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
- MR Author ID: 238389
- Email: yunping.jiang@qc.cuny.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3314077