Harmonic maps for Hitchin representations

  • Qiongling LiEmail author


Let (S,g0) be a hyperbolic surface, \({\rho}\) be a Hitchin representation for \({PSL(n,{\mathbb{R}})}\), and f be the unique \({\rho}\)-equivariant harmonic map from \({({\widetilde{S}}, \widetilde g_0)}\) to the corresponding symmetric space. We show its energy density satisfies \({e(f)\geq 1}\) and equality holds at one point only if \({e(f)\equiv 1}\) and \({\rho}\) is the base \({n}\)-Fuchsian representation of (S,g0). In particular, we show given a Hitchin representation \({\rho}\) for \({PSL(n,{\mathbb{R}})}\), every \({\rho}\)-equivariant minimal immersion f from the hyperbolic plane \({{\mathbb{H}}^2}\) into the corresponding symmetric space X is distance-increasing, i.e. \({f^*g_{X}\geq g_{{\mathbb{H}}^2}}\). Equality holds at one point only if it holds everywhere and \({\rho}\) is an n-Fuchsian representation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author wants to thank the referee for many useful comments and corrections. The author is supported in part by the center of excellence Grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95). The author acknowledges support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). The author acknowledges support from Nankai Zhide Foundation.


  1. Bar10.
    D. Baraglia. \(G_2\) Geometry and Integrable System. Thesis. arXiv:1002.1767v2 (2010)
  2. BD.
    A. Beilinson and V. Drinfeld. Opers. arXiv:math/0501398
  3. BR90.
    F. B. Burstall and J. H. Rawnsley. Twistor Theory for Riemannian Symmetric Spaces. Lecture Notes in Mathematics, Vol. 1424. Springer, Berlin (1990). (With applications to harmonic maps of Riemann surfaces).Google Scholar
  4. Col16.
    B. Collier. Finite Order Automorphism of Higgs Bundles: Theory and Application. Thesis (2016)Google Scholar
  5. CW.
    B. Collier and R. Wentworth. Conformal Limits and the Bialynicki-Birula Stratification of the Space of \(\lambda \)-Connections. arXiv:1808.01622
  6. Cor88.
    Corlette, K.: Flat \(G\)-bundles with canonical metrics. J. Diff. Geom. 28(3), 361–382 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. DL.
    S. Dai and Q. Li. Minimal surfaces for Hitchin representations. To appear in Journal of Differential Geometry, arXiv:1605.09596v2
  8. DL18.
    Dai, S., Li, Q.: On cyclic Higgs bundles. Math. Ann. (2018). Google Scholar
  9. Dal08.
    P. Dalakov, Higgs bundles and opers. Thesis (2008)Google Scholar
  10. DT.
    Deroin, B., Tholozan, N.: Dominating surface group representations by Fuchsian ones. Int. Math. Res. Not. IMRN 13, 4145–4166 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Don87.
    S. K. Donaldson. Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3), (1)55 (1987) 127–131Google Scholar
  12. Hit87.
    N. J. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) (1)55 (1987), 59–126Google Scholar
  13. Hit92.
    Hitchin, N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Jos07.
    J. Jost. Partial differential equations. Second edition. Graduate Texts in Mathematics, 214. Springer, New York, (2007)Google Scholar
  15. Kos59.
    B. Kostant The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81 (1959), 973-1032Google Scholar
  16. Lab06.
    F. Labourie Anosov flows, surface groups and curves in projective space. Invent. Math., 165 (2006), 51–114Google Scholar
  17. Lab08.
    F. Labourie. Cross ratios, Anosov representations and the energy functional on Teichmüller space. Ann. Sci. École. Norm. Sup. (4), (3)41 (2008), 437–469Google Scholar
  18. Lab17.
    F. Labourie. Cyclic surfaces and Hitchin components in rank 2, Ann. of Math. (2), (1)185 (2017), 1–58Google Scholar
  19. Sam78.
    Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. 4, 211–228 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Sim88.
    Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1(4), 867–918 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina
  2. 2.Centre for Quantum Geometry of Moduli Spaces (QGM), Department of MathematicsAarhus UniversityAarhus CDenmark
  3. 3.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations