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Dergiler / International Electronic Journal of Geometry / 2026 / Volume: 14 Issue: 1

On the Moduli Space of Flat Tori Having Unit Area

İsmail Sağlam(Adana Alparslan Türkeş Bilim ve Teknoloji Üniversitesi)0000-0002-1283-6396
Sayfa
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Yayın Tarihi
2021-04-15
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

We show that these two metrics coincide.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Inspiring from Thurston's asymmetric metric on Teichmüller spaces, we define and study a natural (weak) metric on the Teichmüller space of the torus.
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Temel Bulgular & Veri:We prove that this weak metric is indeed a metric: it separates points and it is symmetric.
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Sonuç & Katkı:We show that these two metrics coincide.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:weak metric · asymmetric metric · Teichmüller space

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.

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Bu Makaleye Atıf Yap

Sağlam, İ. (2021). On the Moduli Space of Flat Tori Having Unit Area. International Electronic Journal of Geometry, Volume: 14 Issue: 1. https://doi.org/10.36890/iejg.754478

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Kaynakça (8)

  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).
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  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).