Deformations of harmonic mappings and variation of the energy

Marco Spinaci (1)

1 IF - Institut Fourier

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.

Mots clés: Twisted harmonic maps; Moduli spaces; Higgs bundles; Variations of Hodge structure; Kähler groups; Energy functional

Domaines: Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]; Variables complexes [math.CV]

Marco Spinaci. Deformations of harmonic mappings and variation of the energy. 2013. ⟨hal-00877528v1⟩