Duke Mathematical Journal

Moduli spaces of isoperiodic forms on Riemann surfaces

Curtis T. McMullen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper describes the intrinsic geometry of a leaf A(L) of the absolute period foliation of the Hodge bundle ΩM¯g: its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of A(L) for general g and then turn to a study of the case g=2. In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on A(L)H whose zeros, poles, and exotic trajectories are analyzed in detail.

Article information

Duke Math. J., Volume 163, Number 12 (2014), 2271-2323.

First available in Project Euclid: 15 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F30: Differentials on Riemann surfaces


McMullen, Curtis T. Moduli spaces of isoperiodic forms on Riemann surfaces. Duke Math. J. 163 (2014), no. 12, 2271--2323. doi:10.1215/00127094-2785588. https://projecteuclid.org/euclid.dmj/1410789516

Export citation


  • [1] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887–2073.
  • [2] C. Birkenhake and H. Lange, Complex Abelian Varieties, Grundlehren Math. Wiss. 302, Springer, Berlin, 1992.
  • [3] A. A. Bolibruch, “Inverse monodromy problems of the analytic theory of differential equations” in Mathematical Events of the Twentieth Century, Springer, Berlin, 2006, 49–74.
  • [4] K. Calta and K. Wortman, On unipotent flows in $\mathcal{H}(1,1)$, Ergodic Theory Dynam. Systems 30 (2010), 379–398.
  • [5] M. Kapovich, Periods of abelian differentials and dynamics, preprint, 2000.
  • [6] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631–678.
  • [7] S. Lang, Elliptic Functions, 2nd ed., with an appendix by J. Tate, Grad. Texts in Math. 112, Springer, New York, 1987.
  • [8] E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv. 79 (2004), 471–501.
  • [9] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857–885.
  • [10] C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Math. Ann. 333 (2005), 87–130.
  • [11] C. T. McMullen, Teichmüller curves in genus two: Torsion divisors and ratios of sines, Invent. Math. 165 (2006), 651–672.
  • [12] C. T. McMullen, Dynamics of $\mathrm{SL}_{2}(\mathbf{R})$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), 397–456.
  • [13] C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math. 129 (2007), 183–215.
  • [14] C. T. McMullen, Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS) 15 (2013), 1223–1243.
  • [15] C. T. McMullen, Cascades in the dynamics of measured foliations, preprint, 2012.
  • [16] M. Möller, “Affine groups of flat surfaces” in Handbook of Teichmüller Theory, Vol. II, IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich, 2009, 369–387.
  • [17] M. Möller and D. Zagier, Theta derivatives and Teichmüller curves, in preparation.
  • [18] R. Mukamel, Orbifold points on Teichmüller curves and Jacobians with complex multiplication, preprint, 2012.
  • [19] D. Mumford, Tata Lectures on Theta, I, Progr. Math. 28, Birkhäuser, Boston, 1983.
  • [20] M. Ratner, “Interactions between ergodic theory, Lie groups, and number theory” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 156–182.
  • [21] M. Schmoll, “Spaces of elliptic differentials” in Algebraic and Topological Dynamics, Contemp. Math. 385, Amer. Math. Soc., Providence, 2005, 303–320.
  • [22] G. van der Geer, Hilbert Modular Surfaces, Ergeb. Math. Grenzgeb. (3) 16, Springer, Berlin, 1988.
  • [23] W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553–583.
  • [24] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics, and Geometry, I, Springer, Berlin, 2006, 437–583.