Research Article
BibTex RIS Cite
Year 2021, Volume: 14 Issue: 1, 59 - 65, 15.04.2021
https://doi.org/10.36890/iejg.754478

Abstract

References

  • [1] Belkhirat, A., Papadopoulos, A., Troyanov, M.: Thurston’s weak metric on the Teichmüller space of the torus. Transactions of the American Mathematical Society, 357(8), 3311-3324 (2005).
  • [2] Busemann, H.: Local metric geometry, Trans. Amer. Math. Soc. 56 200-274 (1944).
  • [3] Greenfield, M., Ji, L.: Metrics and compactifications of Teichmüller spaces of flat tori. Preprint arXiv:1903.10655. (2019).
  • [4] Greenfield, M.: A new modular characterization of the hyperbolic plane. Preprint arXiv:1707.00818. (2017).
  • [5] Sağlam, İ.: Complete flat cone metrics on punctured surfaces. Turkish Journal of Mathematics; 43, 813-832 (2019).
  • [6] Thurston, W.P.: Minimal Stretch maps between hyperbolic surfaces. Preprint arxiv:9801039. (1985).
  • [7] Troyanov, M.: Les surfaces euclidienne a singularites coniques, L’Enseign. Math. 32 79–94 (1986).
  • [8] Troyanov, M.: On the moduli space of singular Euclidean surfaces. In: Handbook of Teichmüller Theory, Vol. 1 (European Mathematical Society). 507–540 (2007).

On the Moduli Space of Flat Tori Having Unit Area

Year 2021, Volume: 14 Issue: 1, 59 - 65, 15.04.2021
https://doi.org/10.36890/iejg.754478

Abstract

Inspiring from Thurston's asymmetric metric on Teichmüller spaces, we define and study a natural (weak) metric on the Teichmüller space of the torus. We prove that this weak metric is indeed a metric: it separates points and it is symmetric. We relate this metric with the hyperbolic metric on the upper half-plane. We define another metric which measures how much length of a closed geodesic changes when we deform a flat structure on the torus. We show that these two metrics coincide.

References

  • [1] Belkhirat, A., Papadopoulos, A., Troyanov, M.: Thurston’s weak metric on the Teichmüller space of the torus. Transactions of the American Mathematical Society, 357(8), 3311-3324 (2005).
  • [2] Busemann, H.: Local metric geometry, Trans. Amer. Math. Soc. 56 200-274 (1944).
  • [3] Greenfield, M., Ji, L.: Metrics and compactifications of Teichmüller spaces of flat tori. Preprint arXiv:1903.10655. (2019).
  • [4] Greenfield, M.: A new modular characterization of the hyperbolic plane. Preprint arXiv:1707.00818. (2017).
  • [5] Sağlam, İ.: Complete flat cone metrics on punctured surfaces. Turkish Journal of Mathematics; 43, 813-832 (2019).
  • [6] Thurston, W.P.: Minimal Stretch maps between hyperbolic surfaces. Preprint arxiv:9801039. (1985).
  • [7] Troyanov, M.: Les surfaces euclidienne a singularites coniques, L’Enseign. Math. 32 79–94 (1986).
  • [8] Troyanov, M.: On the moduli space of singular Euclidean surfaces. In: Handbook of Teichmüller Theory, Vol. 1 (European Mathematical Society). 507–540 (2007).
There are 8 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

İsmail Sağlam 0000-0002-1283-6396

Publication Date April 15, 2021
Acceptance Date October 15, 2020
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Sağlam, İ. (2021). On the Moduli Space of Flat Tori Having Unit Area. International Electronic Journal of Geometry, 14(1), 59-65. https://doi.org/10.36890/iejg.754478
AMA Sağlam İ. On the Moduli Space of Flat Tori Having Unit Area. Int. Electron. J. Geom. April 2021;14(1):59-65. doi:10.36890/iejg.754478
Chicago Sağlam, İsmail. “On the Moduli Space of Flat Tori Having Unit Area”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 59-65. https://doi.org/10.36890/iejg.754478.
EndNote Sağlam İ (April 1, 2021) On the Moduli Space of Flat Tori Having Unit Area. International Electronic Journal of Geometry 14 1 59–65.
IEEE İ. Sağlam, “On the Moduli Space of Flat Tori Having Unit Area”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 59–65, 2021, doi: 10.36890/iejg.754478.
ISNAD Sağlam, İsmail. “On the Moduli Space of Flat Tori Having Unit Area”. International Electronic Journal of Geometry 14/1 (April 2021), 59-65. https://doi.org/10.36890/iejg.754478.
JAMA Sağlam İ. On the Moduli Space of Flat Tori Having Unit Area. Int. Electron. J. Geom. 2021;14:59–65.
MLA Sağlam, İsmail. “On the Moduli Space of Flat Tori Having Unit Area”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 59-65, doi:10.36890/iejg.754478.
Vancouver Sağlam İ. On the Moduli Space of Flat Tori Having Unit Area. Int. Electron. J. Geom. 2021;14(1):59-65.