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 . Here we explicitly compute the constant for a configuration of every type. The constant 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.
@article{PMIHES_2003__97__61_0, author = {Eskin, Alex and Masur, Howard and Zorich, Anton}, title = {Moduli spaces of abelian differentials : the principal boundary, counting problems, and the {Siegel-Veech} constants}, journal = {Publications Math\'ematiques de l'IH\'ES}, pages = {61--179}, publisher = {Springer}, volume = {97}, year = {2003}, doi = {10.1007/s10240-003-0015-1}, zbl = {1037.32013}, language = {en}, url = {http://archive.numdam.org/articles/10.1007/s10240-003-0015-1/} }
TY - JOUR AU - Eskin, Alex AU - Masur, Howard AU - Zorich, Anton TI - Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants JO - Publications Mathématiques de l'IHÉS PY - 2003 SP - 61 EP - 179 VL - 97 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-003-0015-1/ DO - 10.1007/s10240-003-0015-1 LA - en ID - PMIHES_2003__97__61_0 ER -
%0 Journal Article %A Eskin, Alex %A Masur, Howard %A Zorich, Anton %T Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants %J Publications Mathématiques de l'IHÉS %D 2003 %P 61-179 %V 97 %I Springer %U http://archive.numdam.org/articles/10.1007/s10240-003-0015-1/ %R 10.1007/s10240-003-0015-1 %G en %F PMIHES_2003__97__61_0
Eskin, Alex; Masur, Howard; Zorich, Anton. Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants. Publications Mathématiques de l'IHÉS, Tome 97 (2003), pp. 61-179. doi : 10.1007/s10240-003-0015-1. http://archive.numdam.org/articles/10.1007/s10240-003-0015-1/
1. Riemann surfaces and spin structures, Ann. Scient. ÉNS 4e Série, 4 (1971), 47-62. | Numdam | MR | Zbl
,2. 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. 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. 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. Billiards in polygons, Physica D, 19 (1986), 311-333. | MR | Zbl
,7. Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., 103 (2) (2000), 191-213. | MR | Zbl
, ,8. Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. | MR | Zbl
, ,9. Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2) (2001), 461-495. | Numdam | MR | Zbl
, ,10. Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. | MR | Zbl
,11. Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760-764. | MR | Zbl
, ,12. Ergodicity of Billiard Flows and Quadratic Differentials, Ann. Math., 124 (1986), 293-311. | MR | Zbl
, , ,13. 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. Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (3) (2003), 631-678. | MR | Zbl
, ,15. Interval exchange transformations and measured foliations, Ann Math., 115 (1982), 169-200. | MR | Zbl
,16. Hausdorff dimension of sets of nonergodic foliations, Ann. Math., 134 (1991), 455-543. | MR | Zbl
, ,17. Flat structures and rational billiards, Handbook on Dynamical systems, Vol. 1A, 1015-1089, North-Holland, Amsterdam 2002. | MR | Zbl
, ,18. Quadratic differentials, Springer 1984. | MR | Zbl
,19. Teichmuller geodesic flow, Ann. Math. 124 (1986), 441-530. | MR | Zbl
,20. Moduli spaces of quadratic differentials, J. D'Analyse Math., 55 (1990), 117-171. | Zbl
,21. Teichmuller curves in moduli space. Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1990), 117-171. | MR
,22. Siegel measures, Ann. Math., 148 (1998), 895-944. | MR | Zbl
,23. 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
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