Duke Mathematical Journal

Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases

Vladimir Markovic

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Tg, the Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential φ. First, we construct a holomorphic embedding E:Hk→Tg,n corresponding to φ. The criterion says that the two metrics agree on this disk if and only if a certain function Φ:E(Hk)→H can be extended to a holomorphic function Φ:Tg,n→H. We then show by explicit computation that this is not the case for quadratic differentials arising from L-shaped pillowcases.

Article information

Source
Duke Math. J., Volume 167, Number 3 (2018), 497-535.

Dates
Received: 19 September 2016
Revised: 31 July 2017
First available in Project Euclid: 24 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1516762971

Digital Object Identifier
doi:10.1215/00127094-2017-0041

Mathematical Reviews number (MathSciNet)
MR3761105

Zentralblatt MATH identifier
06848178

Subjects
Primary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Keywords
Carathéodory metric Teichmüller space L-shaped pillowcases

Citation

Markovic, Vladimir. Carathéodory’s metrics on Teichmüller spaces and $L$ -shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497--535. doi:10.1215/00127094-2017-0041. https://projecteuclid.org/euclid.dmj/1516762971


Export citation

References

  • [1] J. Agler and N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375–403.
  • [2] L. Bers, Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89–126.
  • [3] C. Costara, The symmetrized bidisc and Lempert’s theorem, Bull. Lond. Math. Soc. 36 (2004), 656–662.
  • [4] J. P. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1993.
  • [5] C. J. Earle, “On the Carathéodory metric in Teichmüller spaces” in Discontinuous Groups and Riemann Surfaces (College Park, Md., 1973), Ann. of Math. Stud. 79, Princeton Univ. Press, Princeton, 1974, 99–103.
  • [6] C. J. Earle, L. A. Harris, J. H. Hubbard, and S. Mitra, “Schwarz’s lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds” in Kleinian Groups and Hyperbolic 3-Manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press, Cambridge, 2003, 363–384.
  • [7] C. J. Earle and I. Kra, “On holomorphic mappings between Teichmüller spaces” in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), Academic Press, New York, 1974, 107–124.
  • [8] A. L. Edmonds, J. H. Ewing, and R. S. Kulkarni, Torsion free subgroups of Fuchsian groups and tessellations of surfaces, Invent. Math. 69 (1982), 331–346.
  • [9] F. P. Gardiner, Approximation of infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 282, no. 1 (1984), 367–383.
  • [10] F. P. Gardiner, Carathéodory’s and Kobayashi metrics on Teichmüller space, in preparation.
  • [11] D. Gekhtman, Asymptotics of the translation flow on holomorphic maps out of the poly-plane, preprint, arXiv:1702.02177v3 [math.CV].
    arXiv: 1702.02177v3
  • [12] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221–274.
  • [13] J. A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. (2) 66 (1957), 440–453.
    Mathematical Reviews (MathSciNet): MR90648
    Digital Object Identifier: doi:10.2307/1969901
  • [14] G. Knese, A Schwarz lemma on the polydisk, Proc. Amer. Math. Soc. 135 (2007), 2759–2768.
    Mathematical Reviews (MathSciNet): MR2317950
    Digital Object Identifier: doi:10.1090/S0002-9939-07-08766-7
  • [15] I. Kra, The Carathéodory metric on abelian Teichmüller disks, J. Anal. Math. 40 (1981), 129–143.
    Mathematical Reviews (MathSciNet): MR659787
    Digital Object Identifier: doi:10.1007/BF02790158
  • [16] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427–474.
    Mathematical Reviews (MathSciNet): MR660145
    Digital Object Identifier: doi:10.24033/bsmf.1948
  • [17] J. Liu, On the existence of Jenkins-Strebel differentials, Bull. Lond. Math. Soc. 36 (2004), 365–377.
  • [18] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857–885.
  • [19] H. Royden, “Automorphisms and isometries of Teichmüller space” in Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Stud. 66, Princeton Univ. Press, Princeton, 1971, 369–383.
  • [20] W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, New York, 1969.

Duke University Press

Duke Mathematical Journal

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more