Abstract
In this paper, we study the moduli spaces of flat surfaces with cone singularities verifying the following property: there exists a union of disjoint geodesic tree on the surface such that the complement is a translation surface. Those spaces can be viewed as deformations of the moduli spaces of translation surfaces in the space of flat surfaces. We prove that such spaces are quotients of flat complex affine manifolds by a group acting properly discontinuously, and preserving a parallel volume form. Translation surfaces can be considered as a special case of flat surfaces with erasing forest, in this case, it turns out that our volume form coincides with the usual volume form (which are defined via the period mapping) up to a multiplicative constant. We also prove similar results for the moduli space of flat metric structures on the n-punctured sphere with prescribed cone angles up to homothety. When all the angles are smaller than 2π, it is known (cf. [T]) that this moduli space is a complex hyperbolic orbifold. In this particular case, we prove that our volume form induces a volume form which is equal to the complex hyperbolic volume form up to a multiplicative constant.
Similar content being viewed by others
References
Bavard C., Ghys E. (1992) Polygones du plan et polyèdres hyperboliques. Geom. Dedicata 43(2): 207–224
Bobenko A.I., Springborn B.A. (2007) A discrete Laplace–Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38(4): 740–756
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Birkhäuser (1992).
Eskin A., Masur H. (2001) Asymptotic formulas on flat surfaces. Ergodic Theory Dynm. Syst. 21(2): 443–478
Eskin A., Masur H., Zorich A. (2003) Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants. Publ. Math. Inst. Hautes Études Sci. 97: 61–179
Eskin A., Okounkov A. (2001) Asymptotics of number of branched coverings of a torus and volume of moduli spaces of holomorphic differentials. Invent. Math. 145(1): 59–104
H. Farkas, I. Kra, Riemann Surfaces, second edition. Graduate Texts in Mathematics 71, Springer-Verlag, New York (1992).
F. Gardiner, Teichmüller Theory and Quadratic Differentials, Pure and Applied Mathematics-A Wiley-Interscience series, 1987.
W.M. Goldman, Complex Hyperbolic Geometry, Oxford Mathematical Monographs, Oxford University Press 1999.
Kerckhoff S., Masur H., Smillie J. (1986) Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124: 293–311
M. Kontsevich, Lyapunov exponents and Hodge theory, in “The Mathematical Beauty of Physics” (Saclay, 1996) (in Honor of C. Itzykson) Adv. Ser. Math. Phys. 24, World Sci. Publishing, River Edge, NJ (1997), 318–332.
Kontsevich M., Zorich A. (2003) Connected components of the moduli spaces of Abelian differentials. Invent. Math. 153(3): 631–678
Lanneau E. (2008) Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. ENS (4) 41(1): 1–56
H. Masur, S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A (B. Hasselblatt, A. Katok, eds.), Elsevier Science B.V. (2002), 1015–1089.
Masur H., Zorich A. (2008) Multiple saddle connections on flat surfaces and the boundary principle of the moduli space of quadratic differentials. Geom. Funct. Anal. 18(3): 919–987
W.P. Thurston, Shape of polyhedra and triangulations of the sphere, in “The Epstein Birthday Schrift”, Geom. Topo. Monogr. 1, Geom. Topo. Pub., Coventry (1998), 511–549.
Troyanov M. (1991) Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324(2): 793–821
M. Troyanov, On the moduli space of singular Euclidean surfaces, in “Handbook of Teichm¨uller theory.” Vol. I, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007), 507–540.
Veech W.A. (1990) Moduli spaces of quadratic differentials. Journal d’Analyse Math. 55: 117–171
Veech W.A. (1993) Flat surfaces. Amer. Journal of Math. 115: 589–689
A. Zorich, Flat surfaces, Frontiers in Number Theory, Physics and Geometry 1: On Random Matrices, Zeta Functions and Dynamical Systems, École de Physique des Houches, France, March 9-21 2003, Springer-Verlag (2006).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nguyen, DM. Triangulations and Volume Form on Moduli Spaces of Flat Surfaces. Geom. Funct. Anal. 20, 192–228 (2010). https://doi.org/10.1007/s00039-010-0056-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-010-0056-9