Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow

Eskin, Alex; Kontsevich, Maxim; Zorich, Anton

Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 207-333.

Mots clés : Modulus Space, Vector Bundle, Lyapunov Exponent, Quadratic Differential, Closed Geodesic

[AtEZ] J. Athreya, A. Eskin, and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $𝐂P^1$, pp. 1–55, | arXiv

[Au1] D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, | arXiv

[Au2] D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, | arXiv

[AvVi] Avila, A.; Viana, M. Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture, Acta Math., Volume 198 (2007), pp. 1-56 | DOI | Zbl

[AvMaY1] Avila, A.; Matheus Santos, C.; Yoccoz, J.-C. SL-invariant probability measures on the moduli spaces of translation surfaces are regular, Geom. Funct. Anal. (2013)

[AvMaY2] A. Avila, C. Matheus Santos, and J.-C. Yoccoz, Work in progress.

[Ba1] Bainbridge, M. Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Volume 11 (2007), pp. 1887-2073 | DOI | Zbl

[Ba2] Bainbridge, M. Billiards in L-shaped tables with barriers, Geom. Funct. Anal., Volume 20 (2010), pp. 299-356 | DOI | Zbl

[BeKzh] Belavin, A. A.; Knizhnik, V. G. Algebraic geometry and the geometry of quantum strings, Phys. Lett. B, Volume 168 (1986), pp. 201-206 | DOI | Zbl

[Br] Bers, L. Spaces of degenerating Riemann surfaces, Discontinuous Groups and Riemann Surfaces, Proc. Conf. (1974), pp. 43-55 | Zbl

[BiBo] Bismut, J.-M.; Bost, J.-B. Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math., Volume 165 (1990), pp. 1-103 | DOI | Zbl

[BiGiSo1] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Commun. Math. Phys., Volume 115 (1988), pp. 49-78 | DOI | Zbl

[BiGiSo2] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott–Chern forms, Commun. Math. Phys., Volume 115 (1988), pp. 79-126 | DOI | Zbl

[BiGiSo3] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. III. Quillen metrics and holomorphic determinants, Commun. Math. Phys., Volume 115 (1988), pp. 301-351 | DOI | Zbl

[BwMö] Bouw, I.; Möller, M. Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. Math., Volume 172 (2010), pp. 139-185 | DOI | Zbl

[CaWn] Calta, K.; Wortman, K. On unipotent flows in $\mathscr{H}(1,1)$, Ergod. Theory Dyn. Syst., Volume 30 (2010), pp. 379-398 | DOI | Zbl

[Ch] Chen, D. Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., Volume 228 (2011), pp. 1135-1162 | DOI | Zbl

[ChMö] Chen, D.; Möller, M. Non-varying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., Volume 16 (2012), pp. 2427-2479 | DOI | Zbl

[EM] Eskin, A.; Masur, H. Asymptotic formulas on flat surfaces, Ergod. Theory Dyn. Syst., Volume 21 (2001), pp. 443-478 | DOI | Zbl

[EMz] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,𝐑)$ action on moduli space, | arXiv

[EO] Eskin, A.; Okounkov, A. Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., Volume 145 (2001), pp. 59-104 | DOI | Zbl

[EMSl] Eskin, A.; Masur, H.; Schmoll, M. Billiards in rectangles with barriers, Duke Math. J., Volume 118 (2003), pp. 427-463 | DOI | Zbl

[EMZ] Eskin, A.; Masur, H.; Zorich, A. Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel–Veech constants, Publ. Math. IHÉS, Volume 97 (2003), pp. 61-179 | DOI | Numdam | Zbl

[EMfMr] Eskin, A.; Marklof, J.; Morris, D. Unipotent flows on the space of branched covers of Veech surfaces, Ergod. Theory Dyn. Syst., Volume 26 (2006), pp. 129-162 | DOI | Zbl

[EOPa] Eskin, A.; Okounkov, A.; Pandharipande, R. The theta characteristic of a branched covering, Adv. Math., Volume 217 (2008), pp. 873-888 | DOI | Zbl

[EKZ] Eskin, A.; Kontsevich, M.; Zorich, A. Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011), pp. 319-353 | DOI | Zbl

[EMzRf] A. Eskin, M. Mirzakhani, and K. Rafi, Counting closed geodesics in strata, | arXiv

[EMzMh] A. Eskin, M. Mirzakhani, and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $SL(2,𝐑)$ action on moduli space, | arXiv

[Fay] Fay, J. Kernel functions, analytic torsion, and moduli spaces, Memoirs of the AMS (1992) | Zbl

[Fo1] Forni, G. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math., Volume 155 (2002), pp. 1-103 | DOI | Zbl

[Fo2] Forni, G. On the Lyapunov exponents of the Kontsevich-Zorich cocycle, Handbook of Dynamical Systems (2006), pp. 549-580 | Zbl

[Fo3] Forni, G. A geometric criterion for the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle, J. Mod. Dyn., Volume 5 (2011), pp. 355-395 | DOI | Zbl

[FoMa] G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, pp. 1–8, | arXiv

[FoMaZ1] Forni, G.; Matheus, C.; Zorich, A. Square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011), pp. 285-318 | Zbl

[FoMaZ2] Forni, G.; Matheus, C.; Zorich, A. Lyapunov spectra of covariantly constant subbundles of the Hodge bundle, Ergod. Theory Dyn. Syst. (2012)

[FoMaZ3] G. Forni, C. Matheus, and A. Zorich, Zero Lyapunov exponents of the Hodge bundle, Comment Math. Helv. | arXiv

[GriHt1] 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.

[GriHt2] J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum, | arXiv | Zbl

[GruKr] Grushevsky, S.; Krichever, I. The universal Whitham hierarchy and geometry of the moduli space of pointed Riemann surfaces, Surveys in Differential Geometry, vol. XIV. Geometry of Riemann Surfaces and Their Moduli Spaces (2009), pp. 111-129 | Zbl

[GuJu] Gutkin, E.; Judge, C. Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000), pp. 191-213 | DOI | Zbl

[Hb] 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. | Zbl

[HtLe] Hubert, P.; Lelièvre, S. Prime arithmetic Teichmüller discs in $\mathscr{H}\left(2\right)$, Isr. J. Math., Volume 151 (2006), pp. 281-321 | DOI | Zbl

[JoLu] Jorgenson, J.; Lundelius, R. Continuity of relative hyperbolic spectral theory through metric degeneration, Duke Math. J., Volume 84 (1996), pp. 47-81 | DOI | Zbl

[KhHb] S. Koch and J. Hubbard, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, J. Differ. Geom. | arXiv

[Kk1] Kokotov, A. 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., Volume 364 (2012), pp. 5645-5671 | DOI | Zbl

[Kk2] Kokotov, A. Polyhedral surfaces and determinant of Laplacian, Proc. Am. Math. Soc., Volume 141 (2013), pp. 725-735 | DOI | Zbl

[KkKt1] Kokotov, A.; Korotkin, D. Tau-functions on spaces of Abelian and quadratic differentials and determinants of Laplacians in Strebel metrics of finite volume, Preprint MPI Leipzig, Volume 46 (2004), pp. 1-48

[KkKt2] Kokotov, A.; Korotkin, D. Bergman tau-function: from random matrices and Frobenius manifolds to spaces of quadratic differentials, J. Phys. A, Volume 39 (2006), pp. 8997-9013 | DOI | Zbl

[KkKt3] Kokotov, A.; Korotkin, D. Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula, J. Differ. Geom., Volume 82 (2009), pp. 35-100 | Zbl

[K] Kontsevich, M. Lyapunov exponents and Hodge theory, The Mathematical Beauty of Physics, Saclay, 1996 (in Honor of C. Itzykson) (1997), pp. 318-332 | Zbl

[KZ1] M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, Preprint IHES M/97/13, pp. 1–16, | arXiv

[KZ2] Kontsevich, M.; Zorich, A. Connected components of the moduli spaces of Abelian differentials, Invent. Math., Volume 153 (2003), pp. 631-678 | DOI | Zbl

[KtZg] Korotkin, D.; Zograf, P. Tau function and moduli of differentials, Math. Res. Lett., Volume 18 (2011), pp. 447-458 | DOI | Zbl

[Kn] 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. | Numdam | Zbl

[La1] Lanneau, E. Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv., Volume 79 (2004), pp. 471-501 | DOI | Zbl

[La2] Lanneau, E. Connected components of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Super., Volume 41 (2008), pp. 1-56 | Numdam | Zbl

[Le] Lelièvre, S. Siegel-Veech constants in $\mathscr{H}\left(2\right)$, Geom. Topol., Volume 10 (2006), pp. 1157-1172 | DOI | Zbl

[Lu] Lundelius, R. Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume, Duke Math. J., Volume 71 (1993), pp. 211-242 | DOI | Zbl

[Mk] Maskit, B. Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn., Volume 10 (1985), pp. 381-386 | DOI | Zbl

[M1] Masur, H. The extension of the Weil–Peterson metric to the boundary of Teichmüller space, Duke Math. J., Volume 43 (1976), pp. 623-635 | DOI | Zbl

[M2] Masur, H. Interval exchange transformations and measured foliations, Ann. Math., Volume 115 (1982), pp. 169-200 | DOI | Zbl

[MSm] Masur, H.; Smillie, J. Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv., Volume 68 (1993), pp. 289-307 | Zbl

[MZ] Masur, H.; Zorich, A. Multiple saddle connections on flat surfaces and principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., Volume 18 (2008), pp. 919-987 | DOI | Zbl

[Ma] Matheus Santos, C. Appendix to the paper of G. Forni, a geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle, J. Mod. Dyn., Volume 4 (2010), pp. 453-486 | DOI | Zbl

[MaY] Matheus Santos, C.; Yoccoz, J.-C. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., Volume 5 (2011), pp. 386-395 | Zbl

[MaYZm] C. Matheus Santos, J.-C. Yoccoz, and D. Zmiaikou, Homology of origamis with symmetries, to appear in Ann. Inst. Fourier, 64 (2014), | arXiv | Numdam

[McITa] McIntyre, A.; Takhtajan, L. Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula, Geom. Funct. Anal., Volume 16 (2006), pp. 1291-1323 | DOI | Zbl

[McM] McMullen, C. Dynamics of $SL(2,𝐑)$ over moduli space in genus two, Ann. Math., Volume 165 (2007), pp. 397-456 | DOI | Zbl

[Min] Minsky, Y. Harmonic maps, length, and energy in Teichmüller space, J. Differ. Geom., Volume 35 (1992), pp. 151-217 | Zbl

[Mö] Möller, M. Shimura- and Teichmüller curves, J. Mod. Dyn., Volume 5 (2011), pp. 1-32 | DOI | Zbl

[OsPhSk] Osgood, B.; Phillips, R.; Sarnak, P. Extremals of determinants of Laplacians, J. Funct. Anal., Volume 80 (1988), pp. 148-211 | DOI | Zbl

[Pe] Peters, C. A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann., Volume 268 (1984), pp. 1-20 | DOI | Zbl

[Po1] Polyakov, A. Quantum geometry of bosonic strings, Phys. Lett. B, Volume 103 (1981), pp. 207-210 | DOI

[Po2] Polyakov, A. Quantum geometry of fermionic strings, Phys. Lett. B, Volume 103 (1981), pp. 211-213 | DOI

[Q] Quillen, D. Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., Volume 19 (1985), pp. 31-34 | DOI | Zbl

[Rf1] Rafi, K. A characterization of short curves of a Teichmüller geodesic, Geom. Topol., Volume 9 (2005), pp. 179-202 | DOI | Zbl

[Rf2] Rafi, K. Thick-thin decomposition for quadratic differentials, Math. Res. Lett., Volume 14 (2007), pp. 333-341 | DOI | Zbl

[Rf3] K. Rafi, Hyperbolicity in Teichmüller space, | arXiv

[RySi] Ray, D.; Singer, I. Analytic torsion for complex manifolds, Ann. Math., Volume 98 (1973), pp. 154-177 | DOI | Zbl

[Sm] J. Smillie, in preparation.

[So] 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. | Zbl

[TaZg] Takhtadzhyan, L. A.; Zograf, P. G. The geometry of moduli spaces of vector bundles over a Riemann surface, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 53 (1989), pp. 753-770 | Zbl

[Tr] Treviño, R. On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, Geom. Dedic., Volume 163 (2013), pp. 311-338 | DOI | Zbl

[Ve1] Veech, W. Gauss measures for transformations on the space of interval exchange maps, Ann. Math., Volume 115 (1982), pp. 201-242 | DOI | Zbl

[Ve2] Veech, W. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989), pp. 553-583 | DOI | Zbl

[Ve3] Veech, W. A. Siegel measures, Ann. Math., Volume 148 (1998), pp. 895-944 | DOI | Zbl

[Vb] Vorobets, Ya. Periodic geodesics on generic translation surfaces, Algebraic and Topological Dynamics (2005), pp. 205-258 | Zbl

[Wo1] Wolpert, S. Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math. Phys., Volume 112 (1987), pp. 283-315 | DOI | Zbl

[Wo2] Wolpert, S. The hyperbolic metric and the geometry of the universal curve, J. Differ. Geom., Volume 31 (1990), pp. 417-472 | Zbl

[Wo3] Wolpert, S. Geometry of the Weil–Petersson completion of Teichmüller space, Surveys in Differential Geometry (2003), pp. 357-393 | Zbl

[Wr1] Wright, A. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., Volume 6 (2012), pp. 405-426 | DOI | Zbl

[Wr2] Wright, A. Schwarz triangle mappings and Teichmüller curves: the Veech–Ward–Bouw–Möller curves, Geom. Funct. Anal., Volume 23 (2013), pp. 776-809 | DOI | Zbl

[Z] Zorich, A. Square tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials, Rigidity in Dynamics and Geometry. Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory (2002), pp. 459-471 | Zbl