Geometriae Dedicata

, 122:185 | Cite as

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

  • Steven B. Bradlow
  • Oscar García-Prada
  • Peter B. Gothen
Original Paper


Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant


Hermitian symmetric spaces Higgs bundles 

Mathematics Subject Classification

Primary 14H60 Secondary 57R57 Secondary 58D29 


  1. 1.
    Atiyah M.F. and Bott R. (1982). The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308: 523–615 MathSciNetGoogle Scholar
  2. 2.
    Biswas I. and Ramanan S. (1994). An infinitesimal study of the moduli of Hitchin pairs. J. London Math. Soc. 49(2): 219–231 MATHMathSciNetGoogle Scholar
  3. 3.
    Bradlow S.B., García-Prada O. and Gothen P.B. (2001). Representations of the fundamental group of a surface in PU(p,q) and holomorphic triples. C. R., Acad. Sci. Paris Sér. I Math. 333: 347–352 MATHGoogle Scholar
  4. 4.
    Bradlow S.B., García-Prada O. and Gothen P.B. (2003). Surface group representations and U(p,q)-Higgs bundles. J. Differ. Geom. 64: 111–170 MATHGoogle Scholar
  5. 5.
    Bradlow S.B., García-Prada O. and Mundeti Riera I. (2003). Relative Hitchin-Kobayashi correspondences for principal pairs. Quart. J. Math. 54: 171–208 MATHCrossRefGoogle Scholar
  6. 6.
    Burger, M., Iozzi, A., Labourie, F., Wienhard, A.: Maximal representations of surface groups: Symplectic Anosov structures. (2005), Preprint, arXiv:math.DG/0506079Google Scholar
  7. 7.
    Burger M., Iozzi A. and Wienhard A. (2003). Surface group representations with maximal Toledo invariant. C. R. Math. Acad. Sci. Paris 336(5): 387–390 MATHMathSciNetGoogle Scholar
  8. 8.
    Burger, M., Iozzi, A., Wienhard, A.: Surface group representations with maximal Toledo invariant (2006) Preprint, arXiv:math.DG/0605656v2Google Scholar
  9. 9.
    Choi S. and Goldman W.M. (1993). Convex real projective structures on closed surfaces are closed. Proc. Am. Math. Soc. 118: 657–661 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Choi S. and Goldman W.M. (1997). The classification of real projective structures on compact surfaces. Bull. Am. Math. Soc. (N.S.) 34: 161–171 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Corlette K. (1988). Flat G-bundles with canonical metrics. J. Differ. Geom. 28: 361–382 MATHMathSciNetGoogle Scholar
  12. 12.
    Domic A. and Toledo D. (1987). The Gromov norm of the Kaehler class of symmetric domains. Math. Ann. 276: 425–432 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Donaldson S.K. (1987). Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc 55(3): 127–131 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Faraut J. and Korányi A. (1994). Analysis on Symmetric Cones, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York Google Scholar
  15. 15.
    Frankel T. (1959). Fixed points and torsion on Kähler manifolds. Ann. Math. 2(70): 1–8 MathSciNetCrossRefGoogle Scholar
  16. 16.
    García-Prada, O., Gothen, P.B., Mundet i Riera, I.: Representations of surface groups in Sp \((2n,\mathbb{R})\).(2006) (preliminary version).Google Scholar
  17. 17.
    García-Prada, O., Mundeti Riera, I.: Representations of the fundamental group of a closed oriented surface in \(Sp(4,\mathbb{R})\). Topology 43, 831–855 (2004)Google Scholar
  18. 18.
    Goldman, W.M.: Discontinuous groups and the Euler class. Ph.D. thesis, University of California, Berkeley (1980)Google Scholar
  19. 19.
    Goldman W.M. (1988). Topological components of spaces of representations. Invent. Math. 93: 557–607 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gothen P.B. (2001). Components of spaces of representations and stable triples. Topology 40: 823–850 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Helgason S. (1998). Differential Geometry, Lie groups and Symmetric Spaces, Mathematics, Vol. 80. Academic Press, San Diego Google Scholar
  22. 22.
    Hernández L. (1991). Maximal representations of surface groups in bounded symmetric domains. Trans. Am. Math. Soc. 324: 405–420 MATHCrossRefGoogle Scholar
  23. 23.
    Hitchin N.J. (1987). The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55(3): 59–126 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hitchin N.J. (1992). Lie groups and Teichmüller space. Topology 31: 449–473 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Korányi A. and Wolf J.A. (1965). Realization of hermitian symmetric spaces as generalized half-planes. Ann. of Math. 81(2): 265–288 CrossRefMathSciNetGoogle Scholar
  26. 26.
    Labourie, F.: Anosov flows, surface groups and curves in projective space. (2004) Preprint, arXiv:math.DG/0401230.Google Scholar
  27. 27.
    Li J. (1993). The space of surface group representations. Manuscripta Math. 78: 223–243 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Markman E. and Xia E.Z. (2002). The moduli of flat PU(p,p)-structures with large Toledo invariants. Math. Z. 240: 95–109 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Narasimhan M.S. and Seshadri C.S. (1965). Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82: 540–567 CrossRefMathSciNetGoogle Scholar
  30. 30.
    Ramanathan A. (1975). Stable principal bundles on a compact Riemann surface. Math. Ann. 213: 129–152 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Satake I. (1980). Algebraic Structures of Symmetric Domains, Kanô Memorial Lectures, Vol. 4. Iwanami Shoten, Tokyo Google Scholar
  32. 32.
    Simpson C.T. (1988). Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1: 867–918 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Simpson C.T. (1992). Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75: 5–95 MATHMathSciNetGoogle Scholar
  34. 34.
    Simpson C.T. (1994). Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math., Inst. Hautes Étud. Sci. 79: 47–129 MATHMathSciNetGoogle Scholar
  35. 35.
    Simpson C.T. (1995). Moduli of representations of the fundamental group of a smooth projective variety II. Publ. Math., Inst. Hautes Étud. Sci. 80: 5–79 MATHGoogle Scholar
  36. 36.
    Toledo D. (1989). Representations of surface groups in complex hyperbolic space. J. Differential Geom. 29: 125–133 MATHMathSciNetGoogle Scholar
  37. 37.
    Turaev V.G. (1984). A cocycle of the symplectic first Chern class and the Maslov index. Funct. Anal. Appl. 18: 35–39 MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Uhlenbeck K.K. (1982). Connections with L p bounds on curvature. Comm. Math. Phys. 83: 31–42 MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Welters G.E. (1983). Polarized abelian varieties and the heat equations. Compositio Math. 49: 173–194 MATHMathSciNetGoogle Scholar
  40. 40.
    Wienhard, A: Bounded cohomology and geometry. Ph.D. thesis, Universität Bonn (2005) arXiv:math.DG/0501258Google Scholar
  41. 41.
    Xia E.Z. (2000). The moduli of flat PU(2,1) structures over Riemann surfaces. Pacific Journal of Mathematics 195: 231–256 MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Xia E.Z. (2003). The moduli of flat U(p,1) structures on Riemann surfaces. Geom. Dedicata 97: 33–43 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Steven B. Bradlow
    • 1
  • Oscar García-Prada
    • 2
  • Peter B. Gothen
    • 3
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Departamento de MatemáticasCSICMadridSpain
  3. 3.Departamento de Matemática Pura, Faculdade de CiênciasUniversidade do PortoPortoPortugal

Personalised recommendations