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 $c\xb7\left(\pi {\mathrm{L}}^{2}\right)$. 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.

@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://www.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://www.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://www.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, Volume 97 (2003), pp. 61-179. doi : 10.1007/s10240-003-0015-1. http://www.numdam.org/articles/10.1007/s10240-003-0015-1/

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