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Journals / International Electronic Journal of Geometry / 2021 / Volume: 14 Issue: 2

Classification of Punctures on Complete Flat Surfaces

İsmail Sağlam(Adana Alparslan Türkeş Bilim ve Teknoloji Üniversitesi)0000-0002-1283-6396
Pages
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Publication Date
2021-10-29
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We show that there are uncountably many modification-equivalence classes of punctures on flat surfaces.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We investigate the behavior of a complete flat metric on a surface near a puncture.
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Key Findings & Data:We prove that there are punctures which are not regular if and only if the curvature at the puncture is $4\pi$.
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Conclusion & Impact:We show that there are uncountably many modification-equivalence classes of punctures on flat surfaces.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Flat surface · regular puncture · irregular puncture

Abstract

We investigate the behavior of a complete flat metric on a surface near a puncture. We call a puncture on a flat surface regular if it has a neighborhood which is isometric to that of a point at infinity of a cone. We prove that there are punctures which are not regular if and only if the curvature at the puncture is $4\pi$. We classify irregular punctures of a flat surface up to modification equivalence, where two punctures are called modification-equivalent if they have isometric neighborhoods. We show that there are uncountably many modification-equivalence classes of punctures on flat surfaces.

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Cite This Article

Sağlam, İ. (2021). Classification of Punctures on Complete Flat Surfaces. International Electronic Journal of Geometry, Volume: 14 Issue: 2. https://doi.org/10.36890/iejg.784598

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References (15)

  1. Ambjorn, J., Carfora, M., Marzuoli, A.: The geometry of dynamical triangulations. Lecture Notes in Phys. New Ser. Monogr. 50, Springer- Verlag, Berlin, (1997).
  2. Bavard, C., Ghys, É.: Polygones du plan et polyedres hyperboliques. Geometriae Dedicata. 43(2), 207-224 (1992).
  3. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. USA: American Mathematical Society, (2001).
  4. Carfora, M., Dappiaggi, C., Marzuoli, A.: The modular geometry of random Regge triangulations. Classical Quantum Gravity. 19 (20), 5195–5220 (2002).
  5. Davis, P.: Circulant Matrices.Wiley, New York, (1970).
  6. Delign, P., Mostow, G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publications Mathématiques de l’IHÉS. 63(1), 5-89 (1986).
  7. Farb, B., Margalit, D.: A primer on mapping class groups. USA: Princeton University Press (2012).
  8. Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Springer Science and Business Media (2007).
  9. Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Mathematische Annalen. 293(1), 277-315 (1992).
  10. Masur, H., Tabachnikov, S.: Rational billiards and flat structures. In: Hasselblatt B, Katok A,editors. Handbook of dynamical systems 1. Amsterdam, Netherlands: Elsevier Science, 1015-1089 (2002).
  11. SaĞlam, ˙I.: Complete flat cone metrics on punctured surfaces. Turkish Journal of Mathematics. 43, 813-832 (2019).
  12. Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. Geometry and Topology monographs. 1, 511-549 (1998).
  13. Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Ens. Math. 32, 79-94 (1986).
  14. Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Transactions of the American Mathematical Society. 324(2), 793-821 (1991).
  15. Troyanov, M.: On the moduli space of singular Euclidean surfaces. In: Athanase Papadopoulos, editor. Handbook of Teichmüller Theory 1. Zurich, Switzerland: European Mathematical Society. 507-540 (2007).