My Ph.D. Thesis

This is a "gentle" introduction to the subject. If you want a shorter, all-mathematics abstract, check out the "Deformations of harmonic mappings and variation of the energy" link, as the main results of that paper and the Ph.D. thesis essentially coincide.

Harmonic maps try to answer the following question: Suppose that you are mapping a manifold onto another manifold, $f \colon X \to Y$; imagine the first one as to be "deformable" and the second one as "rigid". Then let the first one adjust on the other, in order to minimize its "stretch" (you can imagine it as made out of rubber). Formally, this minimizes the $L^2$-norm of the differential, that is, its energy, $E(f) \frac{1}{2} \int_X \|df\|^2 \text{Vol}$. The concept of harmonic maps is very flexible (no pun meant) and appears in several places in pure and applied mathematics; it may be worth noting that Pixar uses them in their animation and have some cool animations on the subject.

When minizing, you have of course to fix some data, otherwise the minimizer would always be a trivial (constant) map. Namely, you have to preserve the topology (and only allow families to vary in a continuous way). In my Ph.D. thesis, the precise meaning of this was: Take the universal cover $\tilde X \to X$ and consider maps $f \colon \tilde X \to Y$, that are equivariant with respect to a representation $\rho \colon \pi_1(X) \to \text{Isom}(Y)$ (the group of isometries of $Y$). The fact that for "good" representations $\rho$ harmonic $\rho$-equivariant maps exist is a very nice theorem of Corlette.

The original question was: Suppose now that you deform $\rho$ slightly (i.e. infinitesimally). How does the associated harmonic map(s) vary? The answer, it turns out, can be given and it involves Hodge theory.

Finally, Higgs bundles: How do they enter the picture? This is a subject in complex geometry; one has to work with Kähler manifolds $X$ and $Y$, where harmonic maps have particularly nice properties. Then, given a representation $\rho$ as above and a harmonic map $f$, one can construct a holomorphic bundle with some more structure, which is called a Higgs bundles. These objects play a big role in the very active field of research which aims at understanding moduli space of representations of fundamental groups of Kähler manifolds (or, actually, just Riemann surfaces). Studying deformations of the harmonic maps allows to study how this link between representations and Higgs bundles vary, as well, and to compute the variations of the energy function (which is simply defined as associating to any $\rho$ the energy of any $\rho$-equivariant harmonic map).

Link: For more details on all this, check out the introduction of the Ph.D. thesis! You can find it here!