Advertisement

Mathematische Zeitschrift

, Volume 278, Issue 3–4, pp 617–648 | Cite as

Deformations of twisted harmonic maps and variation of the energy

  • Marco Spinaci
Article

Abstract

We study the deformations of twisted harmonic maps \(f\) with respect to the representation \(\rho \). After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of \(f\) in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional \(E\) coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the “Betti” moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of \(E\) at critical points.

Keywords

Twisted harmonic maps Moduli spaces Higgs bundles  Variations of Hodge structure Kähler groups  Energy functional 

Mathematics Subject Classification (2010)

53C43 14D07 32G13 

Notes

Acknowledgments

This work is based on the author’s Ph.D. thesis at Université Joseph Fourier (Grenoble), which is publicly available at [36]. The author would like to thank his advisor, Philippe Eyssidieux, for introducing him to the subject and for his invaluable support; the referees of his Ph.D. thesis, Olivier Biquard and Domingo Toledo, for their helpful comments; Pierre Py and Alessandro Carlotto for useful remarks and discussions.

References

  1. 1.
    Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Compact Kähler Manifolds, Mathematical Surveys and Monographs, vol. 44. American Mathematical Society, Providence, RI (1996)Google Scholar
  2. 2.
    Bertram, W.: Differential geometry, Lie groups and symmetric spaces over general base fields and rings. Mem. Am. Math. Soc. 900, 202 (2008)MathSciNetGoogle Scholar
  3. 3.
    Bradlow, S.B., García-Prada, O., Gothen, P.B.: Surface group representations and \({\rm U}(p, q)\)-Higgs bundles. J. Differ. Geom. 64(1), 111–170 (2003). http://projecteuclid.org/getRecord?id=euclid.jdg/1090426889
  4. 4.
    Bradlow, S.B., García-Prada, O., Gothen, P.B.: Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces. Geom. Dedicata 122, 185–213 (2006). doi: 10.1007/s10711-007-9127-y CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)Google Scholar
  6. 6.
    Brockett, R., Sussmann, H.: Tangent bundles of homogeneous spaces are homogeneous spaces. Proc. Am. Math. Soc. 35, 550–551 (1972). doi: 10.2307/2037645 CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Burstall, F.E., Rawnsley, J.H.: Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics, vol. 1424. Springer, Berlin (1990). With applications to harmonic maps of Riemann surfacesGoogle Scholar
  8. 8.
    Corlette, K.: Flat \(G\)-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988). http://projecteuclid.org/getRecord?id=euclid.jdg/1214442469
  9. 9.
    Corlette, K.: Rigid representations of Kählerian fundamental groups. J. Differ. Geom. 33(1), 239–252 (1991). http://projecteuclid.org/getRecord?id=euclid.jdg/1214446037
  10. 10.
    Daskalopoulos, G., Dostoglou, S., Wentworth, R.: Character varieties and harmonic maps to \({ R}\)-trees. Math. Res. Lett. 5(4), 523–533 (1998). doi:  10.4310/MRL.1998.v5.n4.a9 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) 55(1), 127–131 (1987). doi: 10.1112/plms/s3-55.1.127 CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis FlahertyGoogle Scholar
  13. 13.
    Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps, CBMS Regional Conference Series in Mathematics, vol. 50. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1983)Google Scholar
  14. 14.
    Eells Jr, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Fornæss, J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248(1), 47–72 (1980). doi: 10.1007/BF01349254 CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    García-Prada, O., Gothen, P.B., Mundet i Riera, I.: Higgs bundles and surface group representations in the real symplectic group. J. Topol. 6(1), 64–118 (2013). doi: 10.1112/jtopol/jts030 CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977). Grundlehren der Mathematischen Wissenschaften, Vol. 224Google Scholar
  18. 18.
    Goldman, W.M., Millson, J.J.: Deformations of flat bundles over Kähler manifolds. In: Geometry and Topology (Athens, Ga., 1985), Lecture Notes in Pure and Appl. Math., vol. 105, pp. 129–145. Dekker, New York (1987)Google Scholar
  19. 19.
    Goldman, W.M., Millson, J.J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Inst. Hautes Études Sci. Publ. Math. 67, 43–96 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Goldman, W.M., Xia, E.Z.: Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. Mem. Am. Math. Soc. 193(904), viii+69 (2008)Google Scholar
  21. 21.
    Hartman, P.: On homotopic harmonic maps. Can. J. Math. 19, 673–687 (1967)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987). doi: 10.1112/plms/s3-55.1.59 CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Hitchin, N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992). doi: 10.1016/0040-9383(92)90044-I CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Karpelevič, F.I.: Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space. Doklady Akad. Nauk SSSR (N.S.) 93, 401–404 (1953)MathSciNetGoogle Scholar
  25. 25.
    Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, pp. 233–243. Springer, Berlin (1979)Google Scholar
  26. 26.
    Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, vol. 31. Princeton University Press, Princeton, NJ (1984)Google Scholar
  27. 27.
    Lin, F.H.: Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. Math. (2) 149(3), 785–829 (1999). doi: 10.2307/121073 CrossRefzbMATHGoogle Scholar
  28. 28.
    Loubeau, E.: Pluriharmonic morphisms. Math. Scand. 84(2), 165–178 (1999)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mazet, E.: La formule de la variation seconde de l’énergie au voisinage d’une application harmonique. J. Differ. Geom. 8, 279–296 (1973)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mostow, G.D.: Some new decomposition theorems for semi-simple groups. Mem. Am. Math. Soc. 1955(14), 31–54 (1955)MathSciNetGoogle Scholar
  31. 31.
    Parreau, A.: Espaces de représentations complètement réductibles. J. Lond. Math. Soc. (2) 83(3), 545–562 (2011). doi: 10.1112/jlms/jdq076 CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Sampson, J.H.: Applications of harmonic maps to Kähler geometry. In: Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984), Contemp. Math., vol. 49, pp. 125–134. Amer. Math. Soc., Providence, RI (1986). doi: 10.1090/conm/049/833809
  33. 33.
    Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)CrossRefzbMATHGoogle Scholar
  34. 34.
    Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I, II. Inst. Hautes Études Sci. Publ. Math. (80), 5–79 (1995) (1994)Google Scholar
  35. 35.
    Siu, Y.T.: The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math. (2) 112(1), 73–111 (1980). doi: 10.2307/1971321 CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Spinaci, M.: Deformations of twisted harmonic maps and variation of the energy (2013). ArXiv:1310.7694
  37. 37.
    Toledo, D.: Hermitian curvature and plurisubharmonicity of energy on Teichmüller space. Geom. Funct. Anal. 22(4), 1015–1032 (2012). doi: 10.1007/s00039-012-0185-4 CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Zucker, S.: Hodge theory with degenerating coefficients: \(L_2\) cohomology in the Poincaré metric. Ann. Math. (2) 109, 415–476 (1979). doi:  10.2307/1971221 CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut Fourier, Grenoble St Martin d’Hères CedexFrance

Personalised recommendations