Mathematische Zeitschrift

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

Deformations of twisted harmonic maps and variation of the energy

  • Marco Spinaci


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.


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

Mathematics Subject Classification (2010)

53C43 14D07 32G13 



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.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

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

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