Classification of Punctures on Complete Flat Surfaces

Year 2021, , 266 - 276, 29.10.2021

İsmail Sağlam

https://doi.org/10.36890/iejg.784598

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.

Keywords

Flat surface, regular puncture, irregular puncture, conical singularities

Project Number

18119001

References

• [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).