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Triangulated Surfaces and Polyhedral Structures

  • Mauro Carfora
  • Annalisa Marzuoli
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 845)

Abstract

In this chapter we introduce the foundational material that will be used in our analysis of triangulated surfaces and of their quantum geometry. We start by recalling the relevant definitions from Piecewise-Linear (PL) geometry, for which we refer freely to Rourke and Sanderson (Introduction to piecewise-linear topology. Springer-Verlag, New York, 1982) , Thurston (Three-dimensional Geometry and Topology, Vol 1. Princeton University Press, Berkeley, 1997). After these introductory remarks we specialize to the case of Euclidean polyhedral surfaces whose geometrical and physical properties will be the subject of the first part of the book. The focus here is on results which are either new or not readily accessible in the standard repertoire. In particular we discuss from an original perspective the structure of the space of all polyhedral surfaces of a given genus and their stable degenerations. This is a rather delicate point which appears in many guises in quantum gravity, and string theory, and which is related to the role that Riemann moduli space plays in these theories. Not surprisingly, the Witten–Kontsevich model by Kontsevitch (Commun Math Phys 147:1–23, 1992) lurks in the background of our analysis, and some of the notions we introduce may well serve for illustrating, from a more elementary point of view, the often deceptive and very technical definitions that characterize this subject. In such a framework, and in the whole landscaping of the space of polyhedral surfaces an important role is played by the conical singularities associated with the Euclidean triangulation of a surface. We provide, in the final part of the chapter, a detailed analysis of the geometry of these singularities. Their relation with Riemann surfaces theory will be fully developed in Chap. 2.

Keywords

Modulus Space Simplicial Complex Polyhedral Cone Euler Class Ribbon Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ambjørn, J., Carfora, M., Marzuoli, A.: The Geometry of Dynamical Triangulations. Lecture Notes in Physics, vol. m50, Springer Verlag, New York (1997)Google Scholar
  2. 2.
    Ambjörn, J., Durhuus, B., Jonsson, T.: “Quantum Geometry”. Cambridge Monograph on Mathematical Physics, Cambridge University Press, Cambridge (1997) Google Scholar
  3. 3.
    Boileau, M., Leeb, B., Porti, J.: Geometrization of 3-dimensional orbifolds. Ann Math 162, 195–290 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, GTM 82. Springer-Verlag, New York (1982)Google Scholar
  5. 5.
    Busemann, H.: Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, vol. 6, Interscience Publisher Inc., New York (1958)Google Scholar
  6. 6.
    Carfora, M., Marzuoli, A.: “Conformal modes in simplicial quantum gravity and the Weil-Petersson volume of moduli space”. Adv. Math. Theor. Phys. 6, 357 (2002) [arXiv:math-ph/0107028]Google Scholar
  7. 7.
    Guo, R., Luo, F.: Rigidity of polyhedral surfaces. Geom. Topol. 13, 1265–1312 (2009). II. arXiv:0711.0766Google Scholar
  8. 8.
    Igusa, K.: Combinatorial miller-morita-mumford classes and witten cycles. Algebraic Geom. Topol. 4, 473–520 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kapovich, M., Millson, J.J.: On the moduli space of a spherical polygonal linkage. Canad. Math. Bull 42(3), 307–320 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kontsevitch, M.: Intersection theory on the moduli space of curves and the matrix airy functions. Commun. Math. Phys. 147, 1–23 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    Leibon, G.: Characterizing the delaunay decompositions of compact hyperbolic surfaces. Geom. Topol. 6, 361–391 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Luo, F.: Rigidity of Polyhedral Surfaces, arXiv:math/0612714Google Scholar
  13. 13.
    Luo, F.: Rigidity of Polyhedral Surfaces, III. arXiv:1010.3284Google Scholar
  14. 14.
    Luo, F.: Variational Principles on Triangulated Surfaces. To Appear in the Handbook of Geometric Analysis arXiv:0803.4232, Google Scholar
  15. 15.
    Mulase, M., Penkava, M.: Ribbon graphs, quadratic differentials on riemann surfaces, and algebraic curves defined over \(\overline{{\mathbb{Q} }}.\) Asian J Math. 2(4), (1998) 875–920. math-ph/9811024 v2 (1998)Google Scholar
  16. 16.
    Mulase, M., Penkava, M.: Periods of strebel differentials and algebraic curves defined over the field of algebraic numbers. Asian J. Math. 6(4), 743–748 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pachner, U.: Konstruktionsmethoden und das kombinatorische homöomorphieproblem für triangulationen kompakter semilinearer mannigfaltigkeiten. Abh. Math. Sem.Univ. Hamburg 57, 69 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pachner, U.: P.L. Homeomorphic manifolds are equivalent by elementary shellings. Europ. J. Combin. 12, 129–145 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rivin, T.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math 139, 553–580 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Springer-Verlag, New York (1982)zbMATHGoogle Scholar
  21. 21.
    Thurston, W.P.: In: Levy, S. (ed.) Three-dimensional Geometry and Topology, vol. 1. Princeton University Press (1997). See also the full set of Lecture Notes (December 1991 Version), electronically available at the Math. Sci. Research Inst. (Berkeley)Google Scholar
  22. 22.
    Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. Geom. Topol. Monog 1, 511 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324, 793 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Troyanov, M.: Les surfaces euclidiennes a’ singularites coniques. L’Enseignment Math. 32, 79 (1986)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Troyanov, M.: On the Moduli Space of Singular Euclidean Surfaces, arXiv:math/0702666v2 [math.DG]Google Scholar
  26. 26.
    Tutte, W.J.: Acensus of planar triangulations. Canad. J. Math. 14, 21–38 (1962)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mauro Carfora
    • 1
  • Annalisa Marzuoli
    • 1
  1. 1.Dipto. Fisica Nucleare e TeoricaIstituto Nazionale di Fisica Nucleare e Teorica, Sez. di Pavia, Università degli Studi di PaviaPaviaItaly

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