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

Alex Eskin; Howard Masur; Anton Zorich

Publications Mathématiques de l'IHÉS (2003)

  • Volume: 97, page 61-179
  • ISSN: 0073-8301

Abstract

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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.

How to cite

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Eskin, Alex, Masur, Howard, and Zorich, Anton. "Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants." Publications Mathématiques de l'IHÉS 97 (2003): 61-179. <http://eudml.org/doc/104192>.

@article{Eskin2003,
abstract = {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.},
author = {Eskin, Alex, Masur, Howard, Zorich, Anton},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {geodesies on flat surfaces; saddle connections; moduli spaces of Abelian differentials; boundary of moduli space; stratification; Siegel-Veech constants; Siegel-Veech formula; strata; thick-thin decomposition; volume estimates; slit construction; stratum interchange; compound surface; number of saddle connections},
language = {eng},
pages = {61-179},
publisher = {Springer},
title = {Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants},
url = {http://eudml.org/doc/104192},
volume = {97},
year = {2003},
}

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
PB - Springer
VL - 97
SP - 61
EP - 179
AB - 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.
LA - eng
KW - geodesies on flat surfaces; saddle connections; moduli spaces of Abelian differentials; boundary of moduli space; stratification; Siegel-Veech constants; Siegel-Veech formula; strata; thick-thin decomposition; volume estimates; slit construction; stratum interchange; compound surface; number of saddle connections
UR - http://eudml.org/doc/104192
ER -

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