Skip to main content
SpringerLink
Log in

On nonuniqueness of geodesics and geodesic disks in the universal asymptotic Teichmüller space

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmüller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in [[μ]] with h ζ *(μ) < h*(μ) for some point ζ ∈ ∂D, then there exist infinitely many geodesic segments joining [[0]] and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [[μ]] in AT(D).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Earle, C. J., Gardiner, F. P., Lakic, N.: Asymptotic Teichmüller space, Part II: The metric structure. Contemp. Math., 256, 187–219 (2004)

    Article  MATH  Google Scholar 

  2. Earle, C. J., Kra, I., Krushkal’, S. L.: Holomorphic motions and Teichmüller spaces. Trans. Amer. Math. Soc., 343, 927–948 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Earle, C. J., Li, Z.: Isometrically ebmbeded polydisks in infinite-dimensional Teichmüller spaces. J. Geom. Anal., 9, 51–71 (1999)

    Article  MATH  Google Scholar 

  4. Fan, J.: On geodesics in asymptotic Teichmüller spaces. Math. Z., 267, 767–779 (2011)

    Article  MATH  Google Scholar 

  5. Fehlman, R.: Ueber extremale quasikonforme abbidungen. Comment. Math. Helv., 56, 558–580 (1981)

    Article  MathSciNet  Google Scholar 

  6. Gardiner, F. P., Lakic, N.: Quasiconformal Teichmüller Theory, American Mathematical Society, New York, 2000

    MATH  Google Scholar 

  7. Li, Z.: Non-uniqueness of geodesics in infinite-dimensional Teichmüller spaces. Complex Var. Theory Appl., 16, 261–272 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, Z.: Non-uniqueness of geodesics in infinite-dimensional Teichmüller spaces (II). Ann. Acad. Sci. Fenn. Math., 18, 355–367 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Li, Z.: A note on geodesics in infinite-dimensional Teichmüller spaces. Ann. Acad. Sci. Fenn. Math., 20, 301–313 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Li, Z.: Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces. Proc. Amer. Math. Soc., 124, 1459–1465 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, Z.: Geodesic discs in Teichmüller space. Sci. China, Ser. A, 48(8), 1075–1082 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, Z.: On the geodesic geometry of infinite-dimensional Teichmuller spaces. Handbook of Teichmuller Theory, IV, 415–437 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Shen, Y. L.: Some remarks on the geodesics in infinite-dimensional Teichmüller spaces. Acta. Math. Sin., 13, 497–502 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shen, Y. L.: On Teichmüller geometry. Complex Var. Theory Appl., 44, 73–83 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tanigawa, H.: Holomorphic families of geodesic discs in infinite-dimensional Teichmüller spaces. Nagoya Math. J., 127, 117–128 (1992)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, 100191, P. R. China

    Yi Qi & Yan Wu

  2. School of Science, Linyi University, Linyi, 276005, P. R. China

    Yan Wu

Authors
  1. Yi Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Wu.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11371045)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, Y., Wu, Y. On nonuniqueness of geodesics and geodesic disks in the universal asymptotic Teichmüller space. Acta. Math. Sin.-English Ser. 33, 201–209 (2017). https://doi.org/10.1007/s10114-016-5399-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-5399-1

Keywords

MR(2010) Subject Classification

Search

Navigation