The Teichmüller geodesic flow and the geometry of the Hodge bundle

Carlos Matheus[1]

  • [1] CNRS - LAGA, UMR 7539, Univ. Paris 13, 99, Av. J.-B. Clément, 93430, Villetaneuse, France

Séminaire de théorie spectrale et géométrie (2010-2011)

  • Volume: 29, page 73-95
  • ISSN: 1624-5458

Abstract

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The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller geodesic flow. These results have been obtained jointly with J.-C. Yoccoz [MY] and G. Forni, A. Zorich [FMZ1], [FMZ2].

How to cite

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Matheus, Carlos. "The Teichmüller geodesic flow and the geometry of the Hodge bundle." Séminaire de théorie spectrale et géométrie 29 (2010-2011): 73-95. <http://eudml.org/doc/219857>.

@article{Matheus2010-2011,
abstract = {The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller geodesic flow. These results have been obtained jointly with J.-C. Yoccoz [MY] and G. Forni, A. Zorich [FMZ1], [FMZ2].},
affiliation = {CNRS - LAGA, UMR 7539, Univ. Paris 13, 99, Av. J.-B. Clément, 93430, Villetaneuse, France},
author = {Matheus, Carlos},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {Teichmüller dynamics; Kontsevich-Zorich cocycle; Geometry of Hodge bundle; Gauss-Manin connection; variations along geodesics; second fundamental form; totally degenerate origamis},
language = {eng},
pages = {73-95},
publisher = {Institut Fourier},
title = {The Teichmüller geodesic flow and the geometry of the Hodge bundle},
url = {http://eudml.org/doc/219857},
volume = {29},
year = {2010-2011},
}

TY - JOUR
AU - Matheus, Carlos
TI - The Teichmüller geodesic flow and the geometry of the Hodge bundle
JO - Séminaire de théorie spectrale et géométrie
PY - 2010-2011
PB - Institut Fourier
VL - 29
SP - 73
EP - 95
AB - The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller geodesic flow. These results have been obtained jointly with J.-C. Yoccoz [MY] and G. Forni, A. Zorich [FMZ1], [FMZ2].
LA - eng
KW - Teichmüller dynamics; Kontsevich-Zorich cocycle; Geometry of Hodge bundle; Gauss-Manin connection; variations along geodesics; second fundamental form; totally degenerate origamis
UR - http://eudml.org/doc/219857
ER -

References

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  2. K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic, Annals of Math. 175, 2012, 835–908, arXiv:1004.5343. Zbl1254.37005MR2993753
  3. A. Eskin, M. Kontsevich, A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, arxiv:1112.5872, 2011, 1–106. MR2820564
  4. —, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5, n. 2, 2011, 319–353. Zbl1254.32019MR2820564
  5. G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Annals of Math., 155, no. 1, 2002, 1–103. Zbl1034.37003MR1888794
  6. —, On the Lyapunov exponents of the Kontsevich–Zorich cocycle, Handbook of Dynamical Systems v. 1B, B. Hasselblatt and A. Katok, eds., Elsevier, 2006, 549–580. Zbl1130.37302MR2186248
  7. G. Forni and C. Matheus, An example of a Teichmuller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, arXiv:0810.0023v1, 2008, 1–8. 
  8. G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5, n. 2, 2011, 285–318. Zbl1276.37021MR2820563
  9. G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, arxiv:1112.0370, 2011, 1–63. Zbl1290.37002MR2820563
  10. G. Forni, C. Matheus and A. Zorich, Zero Lyapunov exponents of the Hodge bundle, arxiv:1201.6075, 2012, 1–39. 
  11. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp. ISBN: 0-471-32792-1 Zbl0836.14001MR507725
  12. J. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1, Teichmüller theory, Matrix Editions, Ithaca, NY, 2006. xx+459 pp. ISBN: 978-0-9715766-2-9; 0-9715766-2-9 Zbl1102.30001MR2245223
  13. F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr. 281:2, 2008, 219–237. Zbl1159.14012MR2387362
  14. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials, Inventiones Mathematicae, 153:3, 2003, 631–678. Zbl1087.32010MR2000471
  15. C. Matheus, Lyapunov spectrum of Kontsevich-Zorich cocycle on the Hodge bundle of square-tiled cyclic covers I, post at the mathematical blog “Disquisitiones Mathematicae” (http://matheuscmss.wordpress.com/) 
  16. 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, n. 3, 2010, 453–486. Zbl1220.37004MR2729331
  17. C. Matheus, J.-C. Yoccoz and D. Zmiaikou, Homology of origamis with symmetries, arxiv:1207.2423, 2012, 1–35. 
  18. M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5, 2011, 1–32, arXiv:math/0501333v2. Zbl1221.14033MR2787595
  19. A. Zorich, Flat surfaces, in collection “Frontiers in Number Theory, Physics and Geometry. Vol. 1: On random matrices, zeta functions and dynamical systems”; Ecole de physique des Houches, France, March 9–21 2003, P. Cartier; B. Julia; P. Moussa; P. Vanhove (Editors), Springer-Verlag, Berlin, 2006, 439–586. Zbl1129.32012MR2261104

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