Skip to main content
SpringerLink
Log in

On Nonuniqueness of Geodesics in Asymptotic Teichmüller Space

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In an infinite-dimensional Teichmüller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichmüller space \(AT(\Delta )\). We introduce the notions of substantial point and non-substantial point in \(AT(\Delta )\). The set of all non-substantial points is open and dense in \(AT(\Delta )\). It is shown that there are infinitely many geodesics joining a non-substantial point to the basepoint. Although we have difficulty in dealing with the substantial points, we give an example to show that there are infinitely many geodesics connecting a certain substantial point and the basepoint. It is also shown that there are always infinitely many straight lines containing two points in \(AT(\Delta )\).

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. Busemann, H.: The Geometry of Geodesics. Academic Press, New York (1955)

    MATH  Google Scholar 

  2. Earle, C.J., Gardiner, F.P., Lakic, N.: Asymptotic Teichmüller Spaces. Part I. The Complex Structure. Contemporary Mathematics, vol. 256, pp. 17–38. American Mathematical Society, Providence (2000)

  3. Earle, C.J., Marković, V., Saric, D.: Barycentric Extension and the Bers Embedding for Asymptotic Teichmüller Spaces. Contemporary Mathematics. American Mathematical Society, Providence (2002)

    Book  MATH  Google Scholar 

  4. Earle, C.J., Gardiner, F.P., Lakic, N.: Asymptotic Teichmüller Spaces. Part II. The Metric Structure. Contemporary Mathematics. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Fehlmann, R., Sakan, K.: On the set of substantial boundary points for extremal quasiconformal mappings. Complex Var. 6, 323–335 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fujikawa, E.: The action of geometric automorphisms of asymptotic Teichmüller spaces. Michigan Math. J. 54, 269–282 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gardiner, F.P.: Teichmüller Theory and Quadratic Differentials. John Wiley & Sons, New York (1987)

    MATH  Google Scholar 

  10. Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüller Theory. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  11. Gardiner, F.P., Sullivan, D.P.: Symmetric structures on a closed curve. Am. J. Math. 114, 683–736 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, Y., Shen, Y.: On angles in Teichmüller spaces. Math. Z. 277, 181–193 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kravetz, S.: On the geometry of Teichmüller spaces and the structure of their modular groups. Ann. Acad. Sci. Fenn. Ser. A 278, 1–35 (1959)

    MathSciNet  MATH  Google Scholar 

  14. Lakic, N.: Substantial boundary points for plane domains and Gardiner’s conjecture. Ann. Acad. Sci. Fenn. Math. 25, 285–306 (2000)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Matsuzaki, K.: Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces. J. Anal. Math. 102, 1–28 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Miyachi, H.: On invariant distances on asymptotic Teichmüller spaces. Proc. Am. Math. Soc. 134, 1917–1925 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. O’Byrne, B.: On Finsler geometry and applications to Teichmüller spaces. Ann. Math. Stud. 66, 317–328 (1971)

    MathSciNet  MATH  Google Scholar 

  21. Reich, E., Strebel, K.: Extremal quasiconformal mappings with given boundary values. Contributions to Analysis. A Collection of Papers Dedicated to Lipman Bers, pp. 375–391. Academic Press, New York (1974)

  22. Reich, E.: Construction of Hamilton sequences for certain Teichmüller mappings. Proc. Am. Math. Soc. 103, 789–796 (1988)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Yao, G.W.: On nonuniqueness of geodesic disks in infinite-dimensional Teichmüller spaces. Monatsh. Math. doi:10.1007/s00605-015-0834-4 (to appear)

  25. Yao, G.W.: Harmonic maps and asymptotic Teichmüller space. Manuscr. Math. 122, 375–389 (2007)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant No. 11271216).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Guowu Yao

Authors
  1. Guowu Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guowu Yao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, G. On Nonuniqueness of Geodesics in Asymptotic Teichmüller Space . J Geom Anal 27, 1445–1467 (2017). https://doi.org/10.1007/s12220-016-9726-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-016-9726-7

Keywords

Mathematics Subject Classification

Search

Navigation