References
J. Athreya, A. Eskin, and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on \({ \mathbf{C}}\!\operatorname {P}^{1}\), pp. 1–55, arXiv:1212.1660, 2012.
D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, arXiv:1205.2359, 2012.
D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, arXiv:1302.0913, 2013.
A. Avila and M. Viana, Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture, Acta Math., 198 (2007), 1–56.
A. Avila, C. Matheus Santos, and J.-C. Yoccoz, SL-invariant probability measures on the moduli spaces of translation surfaces are regular, Geom. Funct. Anal. (2013), doi:10.1007/s00039-013-0244-5, arXiv:1302.4091.
A. Avila, C. Matheus Santos, and J.-C. Yoccoz, Work in progress.
M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887–2073.
M. Bainbridge, Billiards in L-shaped tables with barriers, Geom. Funct. Anal., 20 (2010), 299–356, 20 (2010), 1306.
A. A. Belavin and V. G. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. B, 168 (1986), 201–206.
L. Bers, Spaces of degenerating Riemann surfaces. in Discontinuous Groups and Riemann Surfaces, Proc. Conf., Univ. Maryland, College Park, Md., 1973, Ann. of Math. Studies, vol. 79, pp. 43–55, Princeton Univ. Press, Princeton, 1974.
J.-M. Bismut and J.-B. Bost, Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math., 165 (1990), 1–103.
J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Commun. Math. Phys., 115 (1988), 49–78.
J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott–Chern forms, Commun. Math. Phys., 115 (1988), 79–126.
J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. III. Quillen metrics and holomorphic determinants, Commun. Math. Phys., 115 (1988), 301–351.
I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. Math., 172 (2010), 139–185.
K. Calta and K. Wortman, On unipotent flows in \({\mathcal{H}}(1,1)\), Ergod. Theory Dyn. Syst., 30 (2010), 379–398.
D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135–1162.
D. Chen and M. Möller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427–2479.
A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergod. Theory Dyn. Syst., 21 (2001), 443–478.
A. Eskin and M. Mirzakhani, Invariant and stationary measures for the \(\operatorname {SL}(2,{\mathbf{R}})\) action on moduli space, arXiv:1302.3320, 2013.
A. Eskin and A. Okounkov, Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., 145 (2001), 59–104.
A. Eskin, H. Masur, and M. Schmoll, Billiards in rectangles with barriers, Duke Math. J., 118 (2003), 427–463.
A. Eskin, H. Masur, and A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel–Veech constants, Publ. Math. IHÉS, 97 (2003), 61–179.
A. Eskin, J. Marklof, and D. Morris, Unipotent flows on the space of branched covers of Veech surfaces, Ergod. Theory Dyn. Syst., 26 (2006), 129–162.
A. Eskin, A. Okounkov, and R. Pandharipande, The theta characteristic of a branched covering, Adv. Math., 217 (2008), 873–888.
A. Eskin, M. Kontsevich, and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319–353.
A. Eskin, M. Mirzakhani, and K. Rafi, Counting closed geodesics in strata, arXiv:1206.5574, 2012.
A. Eskin, M. Mirzakhani, and A. Mohammadi, Isolation, equidistribution, and orbit closures for the \(\operatorname {SL}(2,{\mathbf{R}})\) action on moduli space, arXiv:1305.3015, 2013.
J. Fay, Kernel functions, analytic torsion, and moduli spaces, in Memoirs of the AMS, vol. 464, AMS, Providence, 1992.
G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math., 155 (2002), 1–103.
G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in B. Hasselblatt and A. Katok (eds.) Handbook of Dynamical Systems, vol. 1B, pp. 549–580, Elsevier, Amsterdam, 2006.
G. Forni, A geometric criterion for the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle, J. Mod. Dyn., 5 (2011), 355–395.
G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, pp. 1–8, arXiv:0810.0023, 2010.
G. Forni, C. Matheus, and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285–318.
G. Forni, C. Matheus, and A. Zorich, Lyapunov spectra of covariantly constant subbundles of the Hodge bundle, Ergod. Theory Dyn. Syst. (2012), doi:10.1017/etds.2012.148, arXiv:1112.0370.
G. Forni, C. Matheus, and A. Zorich, Zero Lyapunov exponents of the Hodge bundle, arXiv:1201.6075, 2012, 1–39, to appear in Comment Math. Helv.
J. Grivaux and P. Hubert, Exposants de Lyapunov du flot de Teichmüller (d’après Eskin–Kontsevich–Zorich). Séminaire Nicolas Bourbaki, Octobre 2012, Astérisque, to appear.
J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum, arXiv:1307.3481, 2013, pp. 1–13.
S. Grushevsky and I. Krichever, The universal Whitham hierarchy and geometry of the moduli space of pointed Riemann surfaces, in Surveys in Differential Geometry, vol. XIV. Geometry of Riemann Surfaces and Their Moduli Spaces, pp. 111–129, Int. Press, Somerville, 2009.
E. Gutkin and C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., 103 (2000), 191–213.
J. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Teichmüller theory, vol. 1. With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra. With forewords by William Thurston and Clifford Earle. Matrix Editions, Ithaca, 2006.
P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in \({{\mathcal{H}}}(2)\), Isr. J. Math., 151 (2006), 281–321.
J. Jorgenson and R. Lundelius, Continuity of relative hyperbolic spectral theory through metric degeneration, Duke Math. J., 84 (1996), 47–81.
S. Koch and J. Hubbard, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, arXiv:1301.0062, 2013, to appear in J. Differ. Geom.
A. Kokotov, On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials, Trans. Am. Math. Soc., 364 (2012), 5645–5671.
A. Kokotov, Polyhedral surfaces and determinant of Laplacian, Proc. Am. Math. Soc., 141 (2013), 725–735.
A. Kokotov and D. Korotkin, Tau-functions on spaces of Abelian and quadratic differentials and determinants of Laplacians in Strebel metrics of finite volume, Preprint MPI Leipzig, 46 (2004), 1–48.
A. Kokotov and D. Korotkin, Bergman tau-function: from random matrices and Frobenius manifolds to spaces of quadratic differentials, J. Phys. A, 39 (2006), 8997–9013.
A. Kokotov and D. Korotkin, Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula, J. Differ. Geom., 82 (2009), 35–100.
M. Kontsevich, Lyapunov exponents and Hodge theory, in The Mathematical Beauty of Physics, Saclay, 1996 (in Honor of C. Itzykson), Adv. Ser. Math. Phys., vol. 24, pp. 318–332, World Sci. Publishing, River Edge, 1997, .
M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, Preprint IHES M/97/13, pp. 1–16, arXiv:hep-th/9701164.
M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials, Invent. Math., 153 (2003), 631–678.
D. Korotkin and P. Zograf, Tau function and moduli of differentials, Math. Res. Lett., 18 (2011), 447–458.
R. Krikorian, Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich (d’après Forni, Kontsevich, Zorich). Séminaire Bourbaki. Vol. 2003/2004. Astérisque, 299 (2005), Exp. No. 927, vii, 59–93.
E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv., 79 (2004), 471–501.
E. Lanneau, Connected components of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Super., 41 (2008), 1–56.
S. Lelièvre, Siegel-Veech constants in \({\mathcal{H}}(2)\), Geom. Topol., 10 (2006), 1157–1172.
R. Lundelius, Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume, Duke Math. J., 71 (1993), 211–242.
B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn., 10 (1985), 381–386.
H. Masur, The extension of the Weil–Peterson metric to the boundary of Teichmüller space, Duke Math. J., 43 (1976), 623–635.
H. Masur, Interval exchange transformations and measured foliations, Ann. Math., 115 (1982), 169–200.
H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv., 68 (1993), 289–307.
H. Masur and A. Zorich, Multiple saddle connections on flat surfaces and principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., 18 (2008), 919–987.
C. Matheus Santos, Appendix to the paper of G. Forni, a geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle, J. Mod. Dyn., 4 (2010), 453–486.
C. Matheus Santos and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 5 (2011), 386–395.
C. Matheus Santos, J.-C. Yoccoz, and D. Zmiaikou, Homology of origamis with symmetries, to appear in Ann. Inst. Fourier, 64 (2014), arXiv:1207.2423.
A. McIntyre and L. Takhtajan, Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula, Geom. Funct. Anal., 16 (2006), 1291–1323.
C. McMullen, Dynamics of \(\operatorname {SL}(2,{\mathbf{R}})\) over moduli space in genus two, Ann. Math., 165 (2007), 397–456.
Y. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differ. Geom., 35 (1992), 151–217.
M. Möller, Shimura- and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1–32.
B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148–211.
C. Peters, A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann., 268 (1984), 1–20.
A. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B, 103 (1981), 207–210.
A. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B, 103 (1981), 211–213.
D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., 19 (1985), 31–34.
K. Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol., 9 (2005), 179–202.
K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett., 14 (2007), 333–341.
K. Rafi, Hyperbolicity in Teichmüller space, arXiv:1011.6004 [math.GT].
D. Ray and I. Singer, Analytic torsion for complex manifolds, Ann. Math., 98 (1973), 154–177.
J. Smillie, in preparation.
C. Soulé, Lectures on Arakelov geometry. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992.
L. A. Takhtadzhyan and P. G. Zograf, The geometry of moduli spaces of vector bundles over a Riemann surface, Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 753–770 911, (Russian), translation in Math. USSR-Izv. 35 (1990), 83–100.
R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, Geom. Dedic., 163 (2013), 311–338.
W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math., 115 (1982), 201–242.
W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553–583.
W. A. Veech, Siegel measures, Ann. Math., 148 (1998), 895–944.
Ya. Vorobets, Periodic geodesics on generic translation surfaces, in S. Kolyada, Yu. I. Manin and T. Ward (eds.) Algebraic and Topological Dynamics, Contemporary Math., vol. 385, pp. 205–258, Amer. Math. Soc., Providence, 2005.
S. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math. Phys., 112 (1987), 283–315.
S. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differ. Geom., 31 (1990), 417–472.
S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, in Surveys in Differential Geometry, Boston, MA, 2002, vol. VIII, pp. 357–393, Int. Press, Somerville, 2003.
A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., 6 (2012), 405–426.
A. Wright, Schwarz triangle mappings and Teichmüller curves: the Veech–Ward–Bouw–Möller curves, Geom. Funct. Anal., 23 (2013), 776–809.
A. Zorich, Square tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials, in M. Burger and A. Iozzi (eds.) Rigidity in Dynamics and Geometry. Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK, January 5–July 7, 2000, pp. 459–471, Springer, Berlin, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of the first author is partially supported by NSF grant.
Research of the second and of the third authors is partially supported by ANR grant GeoDyM.
Research of the third author is partially supported by IUF.
About this article
Cite this article
Eskin, A., Kontsevich, M. & Zorich, A. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ.math.IHES 120, 207–333 (2014). https://doi.org/10.1007/s10240-013-0060-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10240-013-0060-3