Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants

Eskin, Alex; Masur, Howard; Zorich, Anton

Publications Mathématiques de l'IHÉS, Tome 97 (2003), pp. 61-179.

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 $c·\left(\pi {\mathrm{L}}^2\right)$. 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.

1. M. Atiyah, Riemann surfaces and spin structures, Ann. Scient. ÉNS 4e Série, 4 (1971), 47-62. | Numdam | MR | Zbl

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. | MR | Zbl

3. A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2) (2001), 443-478. | MR | Zbl

4. A. Eskin, A. Zorich, Billiards in rectangular polygons, to appear.

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. | MR | Zbl

6. E. Gutkin, Billiards in polygons, Physica D, 19 (1986), 311-333. | MR | Zbl

7. E. Gutkin, C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., 103 (2) (2000), 191-213. | MR | Zbl

8. J. Hubbard, H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. | MR | Zbl

9. P. Hubert, T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2) (2001), 461-495. | Numdam | MR | Zbl

10. D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. | MR | Zbl

11. A. Katok, A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760-764. | MR | Zbl

12. S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of Billiard Flows and Quadratic Differentials, Ann. Math., 124 (1986), 293-311. | MR | Zbl

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. | MR | Zbl

14. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (3) (2003), 631-678. | MR | Zbl

15. H. Masur, Interval exchange transformations and measured foliations, Ann Math., 115 (1982), 169-200. | MR | Zbl

16. H. Masur, J. Smillie, Hausdorff dimension of sets of nonergodic foliations, Ann. Math., 134 (1991), 455-543. | MR | Zbl

17. H. Masur, S. Tabachnikov, Flat structures and rational billiards, Handbook on Dynamical systems, Vol. 1A, 1015-1089, North-Holland, Amsterdam 2002. | MR | Zbl

18. K. Strebel, Quadratic differentials, Springer 1984. | MR | Zbl

19. W. Veech, Teichmuller geodesic flow, Ann. Math. 124 (1986), 441-530. | MR | Zbl

20. W. Veech, Moduli spaces of quadratic differentials, J. D'Analyse Math., 55 (1990), 117-171. | Zbl

21. W. Veech, Teichmuller curves in moduli space. Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1990), 117-171. | MR

22. W. Veech, Siegel measures, Ann. Math., 148 (1998), 895-944. | MR | Zbl

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. | MR | Zbl