## Duke Mathematical Journal

### Moduli spaces of isoperiodic forms on Riemann surfaces

Curtis T. McMullen

#### Abstract

This paper describes the intrinsic geometry of a leaf $\mathcal{A}(L)$ of the absolute period foliation of the Hodge bundle $\Omega\overline{\mathcal{M}}_{g}$: its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of $\mathcal{A}(L)$ 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 $\mathcal{A}(L)\cong{ \mathbb{H}}$ whose zeros, poles, and exotic trajectories are analyzed in detail.

#### Article information

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

Dates
First available in Project Euclid: 15 September 2014

https://projecteuclid.org/euclid.dmj/1410789516

Digital Object Identifier
doi:10.1215/00127094-2785588

Mathematical Reviews number (MathSciNet)
MR3263035

Zentralblatt MATH identifier
1371.30037

Subjects
Primary: 30F30: Differentials on Riemann surfaces

#### Citation

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

#### References

• [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.