Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Almost-isometry between the Teichmüller metric and the length-spectrum metric on reduced moduli space for surfaces with boundary


Authors: L. Liu, H. Shiga, W. Su and Y. Zhong
Journal: Trans. Amer. Math. Soc. 369 (2017), 6429-6464
MSC (2010): Primary 30F60; Secondary 51M10
DOI: https://doi.org/10.1090/tran/6877
Published electronically: April 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Teichmüller metric and the length-spectrum metric are almost-isometric on moduli space of hyperbolic surfaces with geodesic boundaries whose lengths are bounded above.


References [Enhancements On Off] (What's this?)

  • [1] D. Alessandrini, L. Liu, A. Papadopoulos, and W. Su, The horofunction compactification of Teichmüller spaces of surfaces with boundary, Topology Appl. 208 (2016), 160-191. MR 3506976, https://doi.org/10.1016/j.topol.2016.05.011
  • [2] Lipman Bers, An inequality for Riemann surfaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 87-93. MR 780038
  • [3] Francesco Bonsante, Kirill Krasnov, and Jean-Marc Schlenker, Multi-black holes and earthquakes on Riemann surfaces with boundaries, Int. Math. Res. Not. IMRN 3 (2011), 487-552. MR 2764871, https://doi.org/10.1093/imrn/rnq070
  • [4] Martin Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011), no. 2, 707-733. MR 2800364, https://doi.org/10.2140/gt.2011.15.707
  • [5] Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
  • [6] Jeffrey Danciger, François Guéritaud, and Fanny Kassel, Margulis spacetimes via the arc complex, Invent. Math. 204 (2016), no. 1, 133-193. MR 3480555, https://doi.org/10.1007/s00222-015-0610-z
  • [7] Benson Farb and Howard Masur, Teichmüller geometry of moduli space. II. $ \mathcal {M}(S)$ seen from far away, In the tradition of Ahlfors-Bers. V, Contemp. Math., vol. 510, Amer. Math. Soc., Providence, RI, 2010, pp. 71-79. MR 2581831, https://doi.org/10.1090/conm/510/10019
  • [8] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
  • [9] Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23-41. MR 559474, https://doi.org/10.1016/0040-9383(80)90029-4
  • [10] Enrico Leuzinger, Reduction theory for mapping class groups and applications to moduli spaces, J. Reine Angew. Math. 649 (2010), 11-31. MR 2746464, https://doi.org/10.1515/CRELLE.2010.086
  • [11] Lixin Liu, Athanase Papadopoulos, Weixu Su, and Guillaume Théret, Length spectra and the Teichmüller metric for surfaces with boundary, Monatsh. Math. 161 (2010), no. 3, 295-311. MR 2726215, https://doi.org/10.1007/s00605-009-0145-8
  • [12] Lixin Liu, Athanase Papadopoulos, Weixu Su, and Guillaume Théret, On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 255-274. MR 2643408, https://doi.org/10.5186/aasfm.2010.3515
  • [13] L. Liu and W. Su, Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space, Bull. Lond. Math. Soc. 43 (2011), no. 6, 1181-1190. MR 2861539, https://doi.org/10.1112/blms/bdr052
  • [14] Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386. MR 802500, https://doi.org/10.5186/aasfm.1985.1042
  • [15] Yair N. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996), no. 2, 249-286. MR 1390649, https://doi.org/10.1215/S0012-7094-96-08310-6
  • [16] Maryam Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), no. 1, 179-222. MR 2264808, https://doi.org/10.1007/s00222-006-0013-2
  • [17] Hiroshige Shiga, On a distance defined by the length spectrum of Teichmüller space, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 315-326. MR 1996441
  • [18] Tuomas Sorvali, The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. Ser. A I 526 (1972), 31. MR 0328066
  • [19] Tuomas Sorvali, On Teichmüller spaces of tori, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 1, 7-11. MR 0435389
  • [20] W. P. Thurston, A spine for Teichmüller space, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30F60, 51M10

Retrieve articles in all journals with MSC (2010): 30F60, 51M10


Additional Information

L. Liu
Affiliation: Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, People’s Republic of China
Email: mcsllx@mail.sysu.edu.cn

H. Shiga
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 O-okayama Meguro-ku, Tokyo 158-0001, Japan
Email: shiga@math.titech.ac.jp

W. Su
Affiliation: Department of Mathematics, Fudan University, 200433, Shanghai, People’s Republic of China – and – Shanghai Center for Mathematical Sciences (SCMS), 200433, Shanghai, People’s Republic of China
Email: suwx@fudan.edu.cn

Y. Zhong
Affiliation: Shanghai Center for Mathematical Sciences (SCMS), 200433, Shanghai, People’s Republic of China
Email: zhongyl0430@gmail.com

DOI: https://doi.org/10.1090/tran/6877
Received by editor(s): March 5, 2015
Received by editor(s) in revised form: September 28, 2015
Published electronically: April 7, 2017
Additional Notes: The first and fourth authors were partially supported by NSFC No. 11271378
The second author was partially supported by JSPS KAKENHI Grant No. 16H03933
The third author was partially supported by NSFC Nos. 11671092, 11631010.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society