Deformations of twisted harmonic maps and variation of the energy

This is the main paper taken from the Ph.D. thesis. All the most important theorems proved there are contained here, the proofs are basically the same but drastically shorter. If you want a "gentler" introduction to the subject read the intro under the Ph.D. thesis part, otherwise here's the official abstract:

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.

Link: This is the arXiv link open to everybody, and this is the published version.