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  2020, 16: 255-288. doi: 10.3934/jmd.2020009

Shimura–Teichmüller curves in genus 5

David Aulicino 1, and  Chaya Norton 2,

1. 

Department of Mathematics, Brooklyn College and CUNY Graduate Center, 2900 Bedford Avenue, Brooklyn, NY 11210-2889, USA
2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received  November 18, 2019 Revised  April 29, 2020

Fund Project: Partially supported by the National Science Foundation under Award Nos. DMS - 1738381, DMS - 1600360, and PSC-CUNY Grant # 61639-00 49

We prove that there are no Shimura–Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura–Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura–Teichmüller curves. Moreover, the property of being a Shimura–Teichmüller curve is equivalent to having completely degenerate Kontsevich–Zorich spectrum.

The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.

Keywords: Teichmüller geodesic flow, Kontsevich–Zorich cocycle, Abelian differentials, translation surfaces, Lyapunov exponents, period matrices, variational formulas.
Mathematics Subject Classification: 32G20, 37D40; Secondary: 14H15, 14G35, 37D25.
Citation: David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009
References:
[1]

D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.  doi: 10.4171/CMH/365.  Google Scholar

[2]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.  doi: 10.1017/etds.2016.26.  Google Scholar

[3]

David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5. Google Scholar

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.  Google Scholar

[6]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.  Google Scholar

[7]

J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973.  Google Scholar

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008). Google Scholar

[9]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.  Google Scholar

[10]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[11]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580. doi: 10.1016/S1874-575X(06)80033-0.  Google Scholar

[12]

S. Grushevsky, I. Krichever and C. Norton, Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.  doi: 10.4213/rm9877.  Google Scholar

[13]

X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581. doi: 10.1093/imrn/rny106.  Google Scholar

[14]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.  doi: 10.1002/mana.200510597.  Google Scholar

[15]

H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635.   Google Scholar

[16]

H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.  doi: 10.1215/S0012-7094-86-05319-6.  Google Scholar

[17]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.  doi: 10.3934/jmd.2011.5.1.  Google Scholar

[18]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.  doi: 10.3934/jmd.2010.4.453.  Google Scholar

[19]

Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988. doi: 10.1007/978-94-009-2885-5.  Google Scholar

[20]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.  doi: 10.1007/s00222-010-0236-0.  Google Scholar

[21]

W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  doi: 10.2307/2007091.  Google Scholar

[22]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

[23]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.  doi: 10.1070/RM1996v051n05ABEH002993.  Google Scholar

[24]

A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143.   Google Scholar

show all references

References:
[1]

D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.  doi: 10.4171/CMH/365.  Google Scholar

[2]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.  doi: 10.1017/etds.2016.26.  Google Scholar

[3]

David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5. Google Scholar

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.  Google Scholar

[6]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.  Google Scholar

[7]

J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973.  Google Scholar

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008). Google Scholar

[9]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.  Google Scholar

[10]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[11]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580. doi: 10.1016/S1874-575X(06)80033-0.  Google Scholar

[12]

S. Grushevsky, I. Krichever and C. Norton, Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.  doi: 10.4213/rm9877.  Google Scholar

[13]

X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581. doi: 10.1093/imrn/rny106.  Google Scholar

[14]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.  doi: 10.1002/mana.200510597.  Google Scholar

[15]

H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635.   Google Scholar

[16]

H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.  doi: 10.1215/S0012-7094-86-05319-6.  Google Scholar

[17]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.  doi: 10.3934/jmd.2011.5.1.  Google Scholar

[18]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.  doi: 10.3934/jmd.2010.4.453.  Google Scholar

[19]

Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988. doi: 10.1007/978-94-009-2885-5.  Google Scholar

[20]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.  doi: 10.1007/s00222-010-0236-0.  Google Scholar

[21]

W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  doi: 10.2307/2007091.  Google Scholar

[22]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

[23]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.  doi: 10.1070/RM1996v051n05ABEH002993.  Google Scholar

[24]

A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143.   Google Scholar

Figure 1.  Schematic of the surface coordinates used in the algorithm: Circle vertices represent zeros and square vertices could represent zeros or regular points
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Figure 2.  Proof of Corollary 5.10: The shaded region on the top of $ C_2 $ represents the admissible locations of $ \tau_0 $
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Table 1.  List of Strata Possibly Containing ST-Curves and Values of $ d_{opt} $: Dashed lines separate genus
Stratum $ d_{opt} $
$ \mathscr{H}(1, 1, 1, 1) $ $ 2 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1) $ $ 4 $
$ \mathscr{H}(2, 2, 2) $ $ 3 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1,2) $ $ 36 $
$ \mathscr{H}(1, 1, 1, 1, 2,2) $ $ 18 $
$ \mathscr{H}(1, 1, 2, 2,2) $ $ 12 $
$ \mathscr{H}(2, 2, 2,2) $ $ 9 $
$ \mathscr{H}(1, 1, 1, 1, 1,3) $ $ 16 $
$ \mathscr{H}(1, 1, 3,3) $ $ 8 $
$ \mathscr{H}(1, 1, 1, 1,4) $ $ 10 $
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Stratum $ d_{opt} $
$ \mathscr{H}(1, 1, 1, 1) $ $ 2 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1) $ $ 4 $
$ \mathscr{H}(2, 2, 2) $ $ 3 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1,2) $ $ 36 $
$ \mathscr{H}(1, 1, 1, 1, 2,2) $ $ 18 $
$ \mathscr{H}(1, 1, 2, 2,2) $ $ 12 $
$ \mathscr{H}(2, 2, 2,2) $ $ 9 $
$ \mathscr{H}(1, 1, 1, 1, 1,3) $ $ 16 $
$ \mathscr{H}(1, 1, 3,3) $ $ 8 $
$ \mathscr{H}(1, 1, 1, 1,4) $ $ 10 $
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