# 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

## Access Full Article

top## Abstract

top## How to cite

topEskin, 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 -

## References

top- 1. M. Atiyah, Riemann surfaces and spin structures, Ann. Scient. ÉNS 4e Série, 4 (1971), 47–62. Zbl0212.56402MR286136
- 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. Zbl0194.24701MR253370
- 3. A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2) (2001), 443–478. Zbl1096.37501MR1827113
- 4. A. Eskin, A. Zorich, Billiards in rectangular polygons, to appear. Zbl06546903
- 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. Zbl1019.32014MR1839286
- 6. E. Gutkin, Billiards in polygons, Physica D, 19 (1986), 311–333. Zbl0593.58016MR844706
- 7. E. Gutkin, C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., 103 (2) (2000), 191–213. Zbl0965.30019MR1760625
- 8. J. Hubbard, H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221–274. Zbl0415.30038MR523212
- 9. P. Hubert, T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2) (2001), 461–495. Zbl0985.32008MR1824961
- 10. D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365–373. Zbl0454.57011MR588283
- 11. A. Katok, A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760–764. Zbl0323.58012MR399423
- 12. S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of Billiard Flows and Quadratic Differentials, Ann. Math., 124 (1986), 293–311. Zbl0637.58010MR855297
- 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. Zbl1058.37508MR1490861
- 14. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (3) (2003), 631–678. Zbl1087.32010MR2000471
- 15. H. Masur, Interval exchange transformations and measured foliations, Ann Math., 115 (1982), 169–200. Zbl0497.28012MR644018
- 16. H. Masur, J. Smillie, Hausdorff dimension of sets of nonergodic foliations, Ann. Math., 134 (1991), 455–543. Zbl0774.58024MR1135877
- 17. H. Masur, S. Tabachnikov, Flat structures and rational billiards, Handbook on Dynamical systems, Vol. 1A, 1015–1089, North-Holland, Amsterdam 2002. Zbl1057.37034MR1928530
- 18. K. Strebel, Quadratic differentials, Springer 1984. Zbl0547.30001MR743423
- 19. W. Veech, Teichmuller geodesic flow, Ann. Math. 124 (1986), 441–530. Zbl0658.32016MR866707
- 20. W. Veech, Moduli spaces of quadratic differentials, J. D’Analyse Math., 55 (1990), 117–171. Zbl0722.30032
- 21. W. Veech, Teichmuller curves in moduli space. Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1990), 117–171. MR1005006
- 22. W. Veech, Siegel measures, Ann. Math., 148 (1998), 895–944. Zbl0922.22003MR1670061
- 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. Zbl1038.37015MR1919417

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.