Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

  • Alex EskinEmail author
  • Howard MasurEmail author
  • Anton ZorichEmail author


A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.

We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c·(πL2). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel–Veech formula.

To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.


Modulus Space Riemann Surface Flat Surface Principal Part Closed Geodesic 
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.


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  1. 1.
    M. Atiyah, Riemann surfaces and spin structures, Ann. Scient. ÉNS 4e Série, 4 (1971), 47–62. Google Scholar
  2. 2.
    E. Calabi, An intrinsic characterization of harmonic 1-forms, Global Analysis, Papers in Honor of K. Kodaira, D. C. Spencer and S. Iyanaga (ed.), pp. 101–117, 1969. Google Scholar
  3. 3.
    A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2) (2001), 443–478. Google Scholar
  4. 4.
    A. Eskin, A. Zorich, Billiards in rectangular polygons, to appear. Google Scholar
  5. 5.
    A. Eskin, A. Okounkov, Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., 145 (1) (2001), 59–104. Google Scholar
  6. 6.
    E. Gutkin, Billiards in polygons, Physica D, 19 (1986), 311–333. Google Scholar
  7. 7.
    E. Gutkin, C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., 103 (2) (2000), 191–213. Google Scholar
  8. 8.
    J. Hubbard, H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221–274. Google Scholar
  9. 9.
    P. Hubert, T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2) (2001), 461–495. Google Scholar
  10. 10.
    D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365–373. Google Scholar
  11. 11.
    A. Katok, A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760–764. Google Scholar
  12. 12.
    S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of Billiard Flows and Quadratic Differentials, Ann. Math., 124 (1986), 293–311. Google Scholar
  13. 13.
    M. Kontsevich, Lyapunov exponents and Hodge theory, The mathematical beauty of physics (Saclay, 1996), (in Honor of C. Itzykson) pp. 318–332, Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, 1997. Google Scholar
  14. 14.
    M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (3) (2003), 631–678. Google Scholar
  15. 15.
    H. Masur, Interval exchange transformations and measured foliations, Ann Math., 115 (1982), 169–200. Google Scholar
  16. 16.
    H. Masur, J. Smillie, Hausdorff dimension of sets of nonergodic foliations, Ann. Math., 134 (1991), 455–543. Google Scholar
  17. 17.
    H. Masur, S. Tabachnikov, Flat structures and rational billiards, Handbook on Dynamical systems, Vol. 1A, 1015–1089, North-Holland, Amsterdam 2002. Google Scholar
  18. 18.
    K. Strebel, Quadratic differentials, Springer 1984. Google Scholar
  19. 19.
    W. Veech, Teichmuller geodesic flow, Ann. Math. 124 (1986), 441–530. Google Scholar
  20. 20.
    W. Veech, Moduli spaces of quadratic differentials, J. D’Analyse Math., 55 (1990), 117–171. Google Scholar
  21. 21.
    W. Veech, Teichmuller curves in moduli space. Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1990), 117–171. Google Scholar
  22. 22.
    W. Veech, Siegel measures, Ann. Math., 148 (1998), 895–944. Google Scholar
  23. 23.
    A. Zorich, Square tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials, in collection Rigidity in Dynamics and Geometry, M. Burger, A. Iozzi (eds.), pp. 459–471, Springer 2002.Google Scholar

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© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Institut de MathématiquesUniversité de RennesRennesFrance

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